
497 F.3d 796 (2007)
Pedro Garcia CHAVEZ, also known as Alfredo Garcia Chavez, Petitioner/Appellant,
v.
Douglas L. WEBER, Warden, South Dakota State Penitentiary; Larry Long, Attorney General, State of South Dakota, Respondents/Appellees.
No. 06-2668.
United States Court of Appeals, Eighth Circuit.
Submitted: March 13, 2007.
Filed: August 16, 2007.
*797 *798 Gary G. Colbath, Jr., argued, Assistant Federal Public Defender, Rapid City, South Dakota (Jeffrey L. Viken, on the brief), for appellant.
Frank E. Geaghan, argued, Asst. Attorney General, Pierre, SD (Lawrence E. Long, on the brief), for appellee.
Before WOLLMAN, JOHN R. GIBSON, and MURPHY, Circuit Judges.
JOHN R. GIBSON, Circuit Judge.
Pedro Garcia Chavez was convicted in South Dakota state court of two counts of possession of a controlled substance with intent to distribute and one count of possession of marijuana. He appeals from an order of the district court[1] denying his petition for a writ of habeas corpus filed pursuant to 28 U.S.C. § 2254. We affirm.
We state the facts as found by the state trial court and recited by the Supreme *799 Court of South Dakota in State v. Chavez, 668 N.W.2d 89, 92 (S.D.2003) (affirming Chavez's convictions and sentence on direct appeal). See Bucklew v. Luebbers, 436 F.3d 1010, 1013 (8th Cir.), cert. denied, ___ U.S. ___, 127 S.Ct. 725, 166 L.Ed.2d 565 (2006).
On September 30, 2000, Chavez was driving in Spearfish, South Dakota, when he was pulled over by State Trooper Brian Swets. According to Swets, he initiated the stop because an object was dangling from Chavez's rearview mirror, in violation of South Dakota law.[2]
Swets examined Chavez's driver's license and asked him some general questions. During the questioning, Chavez's front-seat passenger appeared to be unusually nervous and was rolling his jacket in his hands. Swets asked the passenger if he had a green card, and he said that he did not. Swets then asked Chavez if he had a green card, and he said no. Swets asked Chavez, "If I call INS to check, are you going to be here legally?" Chavez said no.
Swets had a drug dog, Crockett, with him on patrol. During his inquiry into Chavez's license, registration, and immigration status, Swets asked Chavez, "[I]f I take my drug dog around the car, is he going to tell me there's drugs in the car?" Chavez shrugged and replied, "[P]robably, if he's trained, probably but I don't think so." Swets had Crockett sniff the car and later testified that Crockett's intense breathing and locked body posture indicated that he detected illegal drugs in the trunk area.
After Crockett's sniff, Swets further investigated Chavez's and his passenger's immigration status. The passenger consented to a search of his wallet, where Swets found what appeared to be a counterfeit social security card. By this time, other officers had arrived on the scene. Swets told the other officers that the passenger had falsified documents and that this created probable cause to search the car.
Swets and other officers then searched Chavez's car, starting at the sunroof after Swets remarked that it appeared to be "out of alignment." They discovered an access panel behind the back seat, which concealed a compartment containing 18.6 pounds of cocaine, 450.1 grams of methamphetamine, and over 4 pounds of marijuana. They seized the drugs, arrested Chavez, and apprised him of his Miranda[3] rights, which he waived.
Chavez was prosecuted initially in federal court for possessing with intent to distribute five kilograms or more of cocaine and for possessing with intent to distribute fifty grams or more of methamphetamine. Chavez moved to suppress the statements he made during the stop and the drugs seized from the car. The evidence before the magistrate judge consisted of Swets's testimony and a videotape from the police car's dashboard camera, which provided an audio recording of the stop and search but no images of Chavez's car because of the way it was positioned relative to Swets's car. The magistrate judge recommended that the motion to suppress be granted, concluding that Swets lacked reasonable suspicion to extend the scope of the traffic stop to inquire about Chavez's immigration status and that Swets failed to read Chavez his Miranda rights before that interrogation. *800 The district court adopted the magistrate judge's recommendation, rejecting the government's argument that the drugs should be admitted because Crockett's sniff of the trunk inevitably would have revealed them. The district court concluded that Crockett never alerted to the presence of drugs based on Swets's statement at the scene. Swets told fellow officers that Crockett "showed interest" and then said that the passenger's falsified documents created probable cause for a search and "that's what [Swets was] going to go ahead with." The court also found it significant that Swets never told his fellow officers that Crockett alerted to the presence of drugs. The government did not appeal the order granting the motion to suppress and dropped the federal charges against Chavez.
About a week after the federal charges were dismissed, Chavez was indicted in South Dakota state court on five drug counts. As he had done in federal court, he filed a motion to suppress statements he made during the search and the seized drugs. The state court reviewed the videotape of the stop and search and heard testimony from Swets. In addition, the State called witnesses who had not testified at the federal court suppression hearing. State Trooper Michael Thomas testified that he was familiar with how Crockett alerts to drugs, that he observed Crockett's sniff of Chavez's car, and that he saw Crockett alert to the presence of drugs in the trunk. Deputy Sheriff Mike Schafer had been on the scene and testified that, when Swets told him that Crockett showed some interest, he interpreted that to mean that Crockett had alerted. The State also called Kyle Heyen, who trained Crockett and many other drug dogs, to testify that sounds of Crockett's breathing on the videotape and the image of Crockett returning to the patrol car after the search with a toy, a reward for an alert, revealed that he had alerted to drugs. The South Dakota state court denied Chavez's motion to suppress, concluding that Crockett had alerted to drugs in the car, providing probable cause for the search and seizure.
In a bench trial, Chavez was convicted of two counts of possession of a controlled substance with intent to distribute and one count of possession of marijuana. He was sentenced to two consecutive terms of ten years' imprisonment on the possession with intent to distribute counts and a concurrent term of ten years on the marijuana count. Chavez appealed to the South Dakota Supreme Court, arguing that the trial court erred by denying his motion to suppress because there was no reasonable suspicion for the stop, the questions about his immigration status exceeded the scope of the traffic stop and violated his Miranda rights, there was no probable cause for the search because Crockett did not alert, and the state court was bound by the federal court's prior determination to that effect. His convictions were upheld in a split decision, with the justices dividing about whether the evidence supported the trial court's finding that Crockett alerted to drugs, providing probable cause for the search. State v. Chavez, 668 N.W.2d 89 (S.D.2003).
Chavez filed an application for a writ of habeas corpus in state court, and the court dismissed it. Chavez filed a motion for issuance of a certificate of probable cause, which also was denied. He filed his petition for writ of habeas corpus with the district court on March 15, 2004. The district court denied the petition, concluding that Stone v. Powell, 428 U.S. 465, 96 S.Ct. 3037, 49 L.Ed.2d 1067 (1976), barred it from reviewing Chavez's Fourth Amendment search and seizure claim, its prior decision suppressing the evidence in Chavez's federal prosecution was not binding on the state court, and any violation of *801 Chavez's Fifth Amendment rights was harmless error. A Certificate of Appealability was granted on June 22, 2006. The issues on appeal are whether Chavez had a full and fair opportunity to litigate his Fourth Amendment claims in the South Dakota courts, whether the doctrine of collateral estoppel compelled the state court to honor the federal district court's suppression of the evidence, and whether any violation of Chavez's Fifth Amendment rights was harmless error.[4]
On an appeal from a district court's denial of a petition for writ of habeas corpus, we review its findings of fact for clear error and its conclusions of law de novo. Lyons v. Luebbers, 403 F.3d 585, 592 (8th Cir.2005). The Antiterrorism and Effective Death Penalty Act provides that, when a claim has been adjudicated on the merits in state court, habeas relief is not warranted unless the state court proceeding "(1) resulted in a decision that was contrary to, or involved an unreasonable application of, clearly established Federal law, as determined by the Supreme Court," or "(2) resulted in a decision that was based on an unreasonable determination of the facts in light of the evidence presented in the State court proceeding." 28 U.S.C. § 2254(d). A state court unreasonably applies clearly established federal law when it "identifies the correct governing legal principle from [the Supreme] Court's decisions but unreasonably applies that principle to the facts of the prisoner's case." Williams v. Taylor, 529 U.S. 362, 413, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000). "[I]t is not enough for us to conclude that, in our independent judgment, we would have applied federal law differently from the state court; the state court's application must have been objectively unreasonable." Rousan v. Roper, 436 F.3d 951, 956 (8th Cir.), cert. denied, ___ U.S. ___, 127 S.Ct. 68, 166 L.Ed.2d 60 (2006). We presume the state court's factual findings to be correct unless Chavez rebuts this presumption with clear and convincing evidence. Rousan, 436 F.3d at 956; 28 U.S.C. § 2254(e)(1).

I.
Chavez first argues that the district court erred in refusing to review his Fourth Amendment claim that the drugs seized from his car should have been suppressed because there was no probable cause for the search. The trial court found that Crockett alerted to the presence of drugs in Chavez's car during his sniff and that this provided probable cause for the subsequent physical search of the car. The South Dakota Supreme Court upheld this conclusion on direct appeal. Chavez raised the issue again on habeas, and the district court concluded that Stone v. Powell, 428 U.S. 465, 96 S.Ct. 3037, 49 L.Ed.2d 1067 (1976), barred it from reviewing Chavez's Fourth Amendment claim. Under Stone, "where the State has provided an opportunity for full and fair litigation of a Fourth Amendment claim, the Constitution does not require that a state prisoner be granted federal habeas *802 corpus relief on the ground that evidence obtained in an unconstitutional search or seizure was introduced at his trial." 428 U.S. at 482, 96 S.Ct. 3037.
Chavez argues that he did not have an opportunity to fully and fairly litigate his Fourth Amendment claim in state court. Chavez contends that the State had an unfair advantage in the prosecution because it was brought on the heels of the failed federal prosecution, which effectively served as a trial run for law enforcement to sharpen the case against him.
The district court correctly concluded that Chavez had an opportunity for full and fair litigation of his Fourth Amendment claim, and thus Stone bars federal habeas review of that claim. To show that he was not afforded an opportunity for full and fair litigation of his claim, Chavez would have to show that the State "provided no corrective procedures at all to address the alleged Fourth Amendment violation" or that the State "provided a corrective mechanism, but [he] was precluded from using that mechanism because of an unconscionable breakdown in the underlying process." Willett v. Lockhart, 37 F.3d 1265, 1271-72 (8th Cir.1994) (en banc) (adopting test set forth in Capellan v. Riley, 975 F.2d 67, 71 (2d Cir.1992)) (quotation marks omitted and emphasis removed); see also Palmer v. Clarke, 408 F.3d 423, 437 (8th Cir.2005). In this case, the State provided a corrective mechanism for any error the trial court may have made in admitting the evidence by allowing Chavez to appeal his convictions to the South Dakota Supreme Court. See Palmer, 408 F.3d at 437. Chavez urges that the Supreme Court was wrong to uphold the trial court's suppression of the evidence. However, a "mere disagreement with the outcome of a state court ruling is not the equivalent of an unconscionable breakdown in the state's corrective process." Capellan, 975 F.2d at 72. Our inquiry focuses on whether Chavez received an opportunity for full and fair litigation of his claim, not on whether legal or factual error in fact occurred. See Willett, 37 F.3d at 1270.
Chavez contends that the successive federal and state litigations of the suppression issue amount to an unconscionable breakdown in the underlying process. We reject his argument. As a threshold matter, we observe that, "although a defendant may not be prosecuted twice by the same sovereign for the same acts, a subsequent prosecution by a separate sovereign does not violate the Constitution." United States v. Johnson, 169 F.3d 1092, 1095-96 (8th Cir.1999). An exception exists where a state prosecution is a "sham" for a federal prosecution, id. at 1096, but the record supports the state courts' conclusion that the federal and state prosecutors acted independently in this case. By calling three witnesses who had not testified at the federal suppression hearing, the State mounted a stronger case against Chavez than the federal government had brought, but this does not amount to an unconscionable breakdown in the system where there is no evidence of control or collusion between the separate prosecuting authorities. See United States v. Davis, 906 F.2d 829, 832-33 (2d Cir. 1990) (suppression issues usually can be separately litigated by separate sovereigns). Chavez contends that the state court's admission of the same evidence that earlier had been suppressed by the federal court contravenes the policy of Stone by allowing one court to second-guess another, but this argument overlooks the fact that the state court analyzed a different record than was before the federal court. Finally, as the district court observed, Chavez was able to appeal these issues to the South Dakota Supreme Court, and he has not argued, nor do we *803 see any evidence of, any breakdown in fairness at the state appellate level. See Willett, 37 F.3d at 1272.
Chavez also contends that Stone does not bar habeas review in his case because Stone applies only to Fourth Amendment searches, and the decisive issue in his claim is whether Crockett alerted to the presence of drugs during his sniff of the car, which is not a Fourth Amendment search according to Illinois v. Caballes, 543 U.S. 405, 408, 125 S.Ct. 834, 160 L.Ed.2d 842 (2005) (holding that a drug dog sniff is not subject to the Fourth Amendment if it is conducted during a traffic stop that is "lawful at its inception and otherwise executed in a reasonable manner"). Chavez also purports to challenge the sniff as unreasonable. These positions cannot coexist. Chavez's argument that the sniff was part of an unreasonably prolonged detention implicates the Fourth Amendment, see Caballes, 543 U.S. at 407, 125 S.Ct. 834, but he declined the opportunity to appeal this issue to the South Dakota Supreme Court, and Stone otherwise bars our review of the state trial court's conclusion that the sniff, which occurred seven minutes into the concededly lawful traffic stop, was reasonable. On the other hand, where a sniff is not a Fourth Amendment search under Caballes, police do not need reasonable suspicion for the sniff, see United States v. Williams, 429 F.3d 767, 772 (8th Cir.2005), and Chavez has no constitutional basis for challenging it. Proceeding from Chavez's position that the dog sniff was not a search, his Fourth Amendment claim is limited to the officers' later physical search of his car, and the sniff is significant only factually as the source of probable cause for that search. Based on the testimony of four witnesses and a review of the videotape recorded at the scene, the state court found that Crockett alerted to the presence of drugs during the sniff and that this provided probable cause for the full search of the car. Though the different judges who have reviewed the videotape and considered the testimony have reached different conclusions, we cannot say that the state court's decision was "based on an unreasonable determination of the facts" in light of the evidence before it, 28 U.S.C. § 2254(d)(2). This is precisely the "grey, twilight area" of Fourth Amendment issues where federal courts do not second-guess state courts on habeas review. Schneckloth v. Bustamonte, 412 U.S. 218, 269, 93 S.Ct. 2041, 36 L.Ed.2d 854 (1973) (Powell, J., concurring). The state courts' conclusion on Chavez's Fourth Amendment claim is classically unreviewable under Stone.

II.
Chavez next argues that his due process rights were violated when the state court denied his motion to suppress after the federal court had ruled in his initial prosecution that the evidence was inadmissible. He argues that the state court was bound by the district court's resolution of the issue under the doctrine of collateral estoppel. The South Dakota Supreme Court rejected this argument after concluding that federal and state officials were not in privity, and the district court concluded that this did not amount to an unreasonable application of clearly established federal law.
The doctrine of collateral estoppel, also known as issue preclusion, provides that "when an issue of ultimate fact has been determined by a valid and final judgment, that issue cannot again be litigated between the same parties in another lawsuit." United States v. Brekke, 97 F.3d 1043, 1049 (8th Cir.1996). The South Dakota Supreme Court concluded that the doctrine did not apply in this case because the State, the party sought to be precluded, was not a party or in privity with a *804 party (the federal government) to the original federal prosecution. See Robinette v. Jones, 476 F.3d 585, 589 (8th Cir.2007) (party to be bound must have been party or in privity with party to earlier suit). This conclusion reasonably applies the correct legal principles and is supported by the record.
The State is bound by the federal court's suppression of the evidence "only if state prosecutors had participated actively in the federal prosecution." Stephens v. Att'y Gen. of Cal., 23 F.3d 248, 249 (9th Cir.1994). The South Dakota courts determined that the State did not actively participate in the federal prosecution of Chavez, and the record supports this finding. Trooper Swets, a state police officer, testified at both suppression hearings, but this does not create privity where the state officers did not have the authority at the federal hearing "to act in the state's name or decide how the prosecution would proceed." Id. There is evidence that federal and state law enforcement authorities cooperated in the case, as the State initially handed the case over to the government because federal drug laws carried the potential for harsher penalties, but this is not out of the ordinary. See United States v. Leathers, 354 F.3d 955, 960 (8th Cir.2004) (cooperation and referral between state and federal authorities is "commonplace and welcome"); see also United States v. Aboumoussallem, 726 F.2d 906, 910 & n. 3 (2d Cir.1984) (on direct appeal of conviction, agreement by one sovereign to prosecute in hopes of imposing harsher penalties does not implicate double jeopardy or due process). In the double jeopardy context,[5] we have held that cooperation between state and federal law enforcement officers "does not in itself affect the identity of the prosecuting sovereign." United States v. Johnson, 169 F.3d 1092, 1096 (8th Cir.1999) (analyzing "sham" exception to dual sovereignty doctrine); see also Bartkus v. Illinois, 359 U.S. 121, 123-24, 79 S.Ct. 676, 3 L.Ed.2d 684 (1959) (critical factor is whether sovereign bringing second prosecution acted independently); United States v. Perchitti, 955 F.2d 674, 676-77 (11th Cir.1992) (double jeopardy cases provide guidance on issue of whether privity exists between two prosecuting sovereigns, and conventional cooperation between state and federal law enforcement does not show privity). Chavez analogizes this case to United States v. Red Bird, 287 F.3d 709, 714 (8th Cir.2002), where, in a federal prosecution, we affirmed the suppression of statements the defendant had made in an interview conducted jointly by FBI and tribal investigators; the defendant had invoked his Sixth Amendment right to counsel in his tribal prosecution and the government and tribe both knew this yet interviewed him without contacting his counsel, id. at 711-12. In contrast to the "close working relationship" between sovereigns in Red Bird, id. at 714, the evidence in this case shows nothing more between the state and federal prosecutors than ordinary cooperation, and this is insufficient to place them in privity.
Courts do not apply collateral estoppel where the doctrine would bar "the United States from using evidence previously suppressed in a state proceeding in which the United States was not a party" or in privity with a party. United States v. Davis, 906 F.2d 829, 832 (2d Cir.1990) (quotation marks and citation omitted); *805 see also Perchitti, 955 F.2d at 675-76; United States v. Safari, 849 F.2d 891, 893 (4th Cir.1988). State courts have applied an analogous rule where, as in this case, a state court admits evidence in a state prosecution after the same evidence was suppressed in an earlier federal prosecution, as long as the state and federal prosecutors acted as separate sovereigns. See, e.g., State v. Brooks, 337 N.C. 132, 446 S.E.2d 579, 588 (1994); People v. Meredith, 11 Cal.App.4th 1548, 15 Cal.Rptr.2d 285, 291-92 (1992). See also Stephens, 23 F.3d at 249 (collateral estoppel did not bar state court from admitting evidence that federal court had suppressed in defendant's earlier federal prosecution where state was neither a party to the federal prosecution nor in privity with the government). We cannot conclude that the South Dakota state courts unreasonably applied clearly established federal law in concluding that collateral estoppel did not require the state trial court to grant Chavez's motion to suppress. See Parker v. Kemna, 260 F.3d 852, 853-54 (8th Cir.2001) (stating standard).

III.
Finally, Chavez argues that the statements he made during the search should not have been admitted because he had not been apprised of his Miranda rights. Chavez advanced the same argument before the South Dakota Supreme Court, which declined to reach the issue in light of its conclusion that the drug dog alerted, providing probable cause for the search of the car and the discovery of drugs that supported Chavez's conviction independent of any incriminating statements he made without the benefit of Miranda. The district court concluded that the admission of the statements was at worst harmless error because, even if the statements had been suppressed, there was overwhelming evidence of Chavez's guilt in the large quantity of drugs found in the hidden compartment in the backseat. See Lyons v. Luebbers, 403 F.3d 585, 597 (8th Cir.2005). An evidentiary error is harmless if it did not affect Chavez's substantial rights and did not influence or had only a slight influence on the verdict. United States v. Looking Cloud, 419 F.3d 781, 787 (8th Cir.2005). "The admission of statements obtained in violation of Miranda may constitute harmless error where there remains overwhelming independent evidence as to the defendant's guilt." United States v. Packer, 730 F.2d 1151, 1157 (8th Cir.1984). The drugs found in Chavez's car provided overwhelming evidence of guilt, and it was Crockett's alert, not Chavez's statements, that led the police to discover them. The district court correctly concluded that the admission of Chavez's incriminating statements, most of which involved his immigration status rather than the drug crimes ultimately charged, was at worst harmless error under these circumstances and that the South Dakota courts reasonably applied federal law in resolving this issue.
We affirm the dismissal of Chavez's petition for habeas relief.
NOTES
[1]  The Honorable Karen E. Schreier, Chief Judge, United States District Court for the District of South Dakota.
[2]  S.D. Codified Law § 32-15-6 provides: "It is a petty offense for any person to drive any vehicle upon a highway with any object or gadget dangling between the view of the driver and the windshield of the vehicle." The statute was amended in 2004 so that enforcement of this law "shall be accomplished as a secondary enforcement."
[3]  Miranda v. Arizona, 384 U.S. 436, 86 S.Ct. 1602, 16 L.Ed.2d 694 (1966).
[4]  Chavez argued before the state courts that the traffic stop was illegal because Swets was on an improper "fishing expedition of a non-Caucasian," as demonstrated by Swets's remark on the audio tape, "What the hell, I'm going to stop them anyway," and Swets's testimony that he inquires further in traffic stops when the driver and passengers speak broken English. The South Dakota Supreme Court concluded that the stop was objectively reasonable, citing United States v. Bloomfield, 40 F.3d 910, 915 (8th Cir.1994) (stating that any traffic violation, "however minor," provides probable cause for a stop, and, if a traffic violation has occurred, an officer's underlying motivations will not render the stop pretextual). In his habeas petition Chavez does not contest that the dangling object justified the traffic stop, so we do not inquire into Swets's possible misconduct.
[5]  Chavez concedes that the successive federal and state prosecutions for the same offense did not violate his Fifth Amendment double jeopardy rights. Collateral estoppel is a separate issue from double jeopardy. See United States ex rel. DiGiangiemo v. Regan, 528 F.2d 1262, 1265 (2d Cir.1975) (collateral estoppel issue can arise even where double jeopardy is not implicated).
