
553 F.3d 1028 (2009)
Karl HARRIS, Petitioner-Appellee/Cross-Appellant,
v.
Blaine C. LAFLER, Respondent-Appellant/Cross-Appellee.
Nos. 05-2104, 05-2159.
United States Court of Appeals, Sixth Circuit.
Argued: December 11, 2008.
Decided and Filed: January 30, 2009.
*1029 ARGUED: Janet A. VanCleve, Office of the Michigan Attorney General, Lansing, Michigan, for Appellant. John F. Royal, Detroit, Michigan, for Appellee. ON BRIEF: B. Eric Restuccia, Office of the Michigan Attorney General, Lansing, *1030 Michigan, for Appellant. John F. Royal, Detroit, Michigan, for Appellee.
Before: ROGERS, SUTTON and McKEAGUE, Circuit Judges.

OPINION
SUTTON, Circuit Judge.
A jury convicted Karl Harris of violating several Michigan criminal laws, including second-degree murder, and the judge sentenced him to 52 to 77 years in prison. His state-court efforts to obtain relief came to naught, and he filed this federal habeas corpus petition, which the district court rejected in all respects, save one: It granted relief on his Brady claim that the prosecution failed to disclose three statements made by the police to the State's lead witness before he testified. We affirm.

I.
On the night of March 8, 1997, Harris and his friend Richard Ward were involved in a fight at a Detroit-area nightclub. What began as a tussle between Harris and another individual, Erwin Smith, escalated into a club-wide melee, which security guards eventually brought to an end with the aid of pepper spray. After the club emptied out, Smith and six of his friends piled into an SUV and drove away, heading east on I-96. Before long, Smith noticed that he was being followed, and, not long after that, someone inside the pursuing vehicle fired several rounds from an AK-47 assault rifle at the SUV. Two passengers in the SUV were killed, and several others were wounded. No one in the SUV could identify the gunman.
The police arrested Ward and Harris about a month after the shooting. Ward gave two statements to the police about the incident and testified at Harris's preliminary examination, stating that he drove the pursuing vehicle on the night of the shooting and that Harris fired the AK-47 at the SUV. (Although Ward is paralyzed from the waist down, he has the capacity to drive a car with hand-held controls.) At Harris's jury trial, Ward invoked his Fifth Amendment rights, after which the trial court allowed the State to introduce Ward's preliminary-examination testimony. Harris, for his part, did not testify, but he contradicted Ward's account of the evening through the testimony of his girlfriend, who said that she was with Harris immediately after the fight at the club and that the two of them spent the remainder of the night at Harris's mother's house. The jury convicted Harris on all eight counts: two counts of second-degree murder, five counts of assault with intent to murder and one count of possession of a firearm during the commission of a felony.
When Harris filed a motion for a new trial, the trial court granted him an evidentiary hearing, where three pieces of information came to light. First, Ward testified that, after the police officers arrested him, they told him that if he gave a statement about the shooting they would release Ward's girlfriend, who had been arrested along with Harris and Ward. Ward gave a statement, but the officers did not release his girlfriend because they "didn't like the statement." JA 818. Ward admitted that he had not told "the complete truth" in the first statement, then offered a second statement two days later, JA 99, after which the police released his girlfriend. The key difference between the two statements was that Ward first claimed that someone else was driving the vehicle on the night of the shooting and that he and Harris were passengers, but he later admitted that he was the driver and that Harris was the sole passenger. Ward pegged Harris as the shooter in both statements.
Second, Ward testified that, on the day of Harris's preliminary examination, a police *1031 officer told Ward that if he testified consistently with his second statement the police would release him. At the preliminary examination, Ward testified consistently with his second statement, and the police released him later that day.
Third, the same officer told Ward that if anyone asked him whether he had been promised anything in exchange for his testimony, he should deny that any promises were made. When Ward testified at the preliminary examination, Harris's counsel asked him several times whether the police had promised him anything in exchange for his testimony. Ward denied that any promises had been made.
Notwithstanding this evidence, the trial court denied Harris's motion for a new trial, including his claim that the State violated his due-process rights by failing to produce exculpatory evidence. The Michigan Court of Appeals affirmed his conviction, and the Michigan Supreme Court denied review.
When Harris filed a federal habeas corpus petition, the district court sized up his due-process claim differently. Relying on Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), and its heirs, the court concluded that the prosecutor's failure to disclose these three exchanges between Ward and the police violated Harris's due-process rights. The district court denied Harris's other claims. The warden appealed the district court's grant of the writ, and Harris, after receiving a certificate of appealability, cross-appealed the denial of three of his other claims.

II.
Before addressing the merits of the district court's decision, we must resolve a threshold question of process. A federal district court, generally speaking, may not grant the writ on a "mixed" petition, one containing claims that the petitioner has pressed before the state courts and claims that he has not. See 28 U.S.C. § 2254(b)(1)(A); Rhines v. Weber, 544 U.S. 269, 273-74, 125 S.Ct. 1528, 161 L.Ed.2d 440 (2005).
At the district court, the State contended that Harris had not exhausted three of his thirteen claims in the state courts and contended that the district court therefore had no authority to address the merits of any of Harris's claims, including his exhausted Brady claim. Harris, as it turns out, raised one of these claims in the state courts, namely his claim that the courts should have immunized Ward so that he could testify in person at Harris's trial. But the State was correct that Harris did not "fairly present" the other two claims to the state courts: that the trial judge took inadequate steps to protect the jury from being tainted by threatening comments made to two jurors during a trial recess, and that he improperly led the jury to believe that it could not have trial testimony read back to it during deliberations. See Baldwin v. Reese, 541 U.S. 27, 29, 124 S.Ct. 1347, 158 L.Ed.2d 64 (2004). For reasons never explained, the district court did not address the State's argument that the court lacked authority to address any of Harris's claims until they had all been exhausted or voluntarily dismissed. The court instead proceeded to address the merits of each of Harris's arguments and granted relief on one of them. That was error. See Rhines, 544 U.S. at 273-74, 125 S.Ct. 1528; 28 U.S.C. § 2254(b)(1)(A).
The question is what to do about it. When faced with this predicament in the past, we have vacated the order granting the writ and remanded the case to the district court so that it could do one of four things: (1) dismiss the mixed petition in its entirety, Rhines, 544 U.S. at 274, 125 S.Ct. 1528; (2) stay the petition and hold it in abeyance while the petitioner returns to state court to raise his unexhausted claims, *1032 id. at 275, 125 S.Ct. 1528; (3) permit the petitioner to dismiss the unexhausted claims and proceed with the exhausted claims, id. at 278, 125 S.Ct. 1528; or (4) ignore the exhaustion requirement altogether and deny the petition on the merits if none of the petitioner's claims has any merit, 28 U.S.C. § 2254(b)(2). See Rockwell v. Yukins, 217 F.3d 421, 425 (6th Cir.2000).
That path normally is the correct one to take, and we would take it here but for the fact that this case presents two additional wrinkles: Harris has agreed in his appellate briefs (and at oral argument) to dismiss with prejudice any unexhausted claims, and the State in response has not insisted that we remand the case to the district court before we address the merits of the exhausted claims. Nothing prevents Harris (through his counsel) from agreeing to dismiss some of his claims with prejudice, and nothing prevents us from holding him to his promise. Nor does anything prevent the State from agreeing to allow us to review the exhausted claims or for that matter the unexhausted claims at this point. The exhaustion requirement does not define the subject-matter jurisdiction of the federal courts. See, e.g., White v. Mitchell, 431 F.3d 517, 526 (6th Cir. 2005). The requirement is designed to give the States and the state courts a first look at a habeas petitioner's claims, Rhines, 544 U.S. at 273, 125 S.Ct. 1528, and a State is free to waive that right when it wishes, see 28 U.S.C. § 2254(b)(3); D'Ambrosio v. Bagley, 527 F.3d 489, 495 (6th Cir.2008).
In the unusual setting of this case where the district court apparently overlooked unexhausted claims and the State on appeal has identified no interest supporting a remand for dismissal of those claims or for that matter urged us to remand the case for that purposethere is good reason for permitting the parties to move immediately to the substance of the dispute at the expense of its traditional form. As the district court's decision suggests, Harris's Brady claim is a serious one. If Harris and the district court are correct about the merits of this claim, he is entitled to a new trial, one that may give him what he wantsa release from custodya form of relief he no doubt wishes to secure sooner rather than later. The State may perceive that its interests cut in the same direction. If Harris and the district court are correct, the State has no interest in delaying that new trial, as time runs the risk of making any evidence of guilt stale, if not non-existent. And if Harris and the district court are wrong, the State has an interest in removing the threat of a new trial now rather than later. The alternativevacating the district court's decision, remanding the case to the district court, permitting it to dismiss Harris's two unexhausted claims, allowing it to re-enter its same order, then permitting the State to appeal againthus serves no one in this instance and neither party has argued otherwise. We therefore will permit Harris to abandon claims ten and eleven, will proceed to the merits of Harris's Brady claim and will direct the district court on remand to dismiss claims ten and eleven of Harris's habeas petition-his two unexhausted (and now voluntarily abandoned) claims.

III.
Because Harris filed his habeas petition after the effective date of the Antiterrorism and Effective Death Penalty Act of 1996 (AEPDA), we may grant relief on Harris's Brady claim only if the state-court rulings were "contrary to, or involved an unreasonable application of, clearly established Federal law, as determined by the Supreme Court of the United States." 28 U.S.C. § 2254(d). To obtain relief on his Brady claim, Harris must *1033 establish that: (1) the State suppressed relevant evidence; (2) the evidence was favorable to him; and (3) there is a "reasonable probability" that the outcome of the trial would have been different if the evidence had been disclosed. See Strickler v. Greene, 527 U.S. 263, 281-82, 289-90, 119 S.Ct. 1936, 144 L.Ed.2d 286 (1999) (internal quotation marks omitted). The question, then, is whether the state courts contradicted or unreasonably applied these requirements in rejecting Harris's Brady claim.
There is not much to say about the first two requirements. As to the first, the police made three pertinent statements to Ward: that his girlfriend would be released if they were satisfied with his statement about the incident; that Ward would be released if he testified at Harris's preliminary examination consistently with his second statement to the police; and that Ward should not tell anyone that the police had promised him anything in return for his statements or testimony. The prosecution did not disclose these statements, and they plainly were relevant to the credibility of Ward's testimony. Contrary to the state court of appeals' assumption, People v. Harris, No. 212870, 2001 WL 1004367, at *1 (Mich.Ct.App. Aug.31, 2001), it makes no difference that the prosecutors did not know about the police officers' statements. The Brady obligation applies even to "evidence known only to police investigators" and thus imposes upon prosecutors "a duty to learn of any favorable evidence known to the others acting on the government's behalf ... including the police." Strickler, 527 U.S. at 280-81, 119 S.Ct. 1936 (internal quotation marks omitted). Brady thus applies to relevant evidence in the hands of the police, whether the prosecutors knew about it or not, whether they suppressed it intentionally or not, id., and whether the accused asked for it or not, id. at 280, 119 S.Ct. 1936; accord United States v. Agurs, 427 U.S. 97, 107, 96 S.Ct. 2392, 49 L.Ed.2d 342 (1976).
As to the second requirement, these statements amount to favorable evidence. Because Brady applies not only to exculpatory evidence but also to impeachment evidence, see United States v. Bagley, 473 U.S. 667, 676, 105 S.Ct. 3375, 87 L.Ed.2d 481 (1985), and because Harris could have used these statements to cast doubt on the credibility of Ward's testimony, Brady covers the statements. Harris, then, satisfies the first two elements of a Brady claim, and indeed the State does not argue otherwise.
There is more to say about the third requirement, prejudice, and the state courts' apparent conclusion"apparent" because the state courts never specifically said anything about this prong of the testthat the disclosure of this evidence would not have created a reasonable probability that the outcome at trial would have been different. The district court held that the state courts unreasonably applied this requirement, and so do we.
Ward, to begin, was not a run-of-the-mill witness. He was the key witness for the prosecution. Ward gave an eyewitness, on-the-scene account of the shooting that explicitly identified Harris as the gunman. And this testimony not only implicated Harris, but it also implicated Ward as an accomplice in the murder, giving the testimony a highly credible veneer.
Nor was Ward's testimony one damming piece of evidence among many; it was the only piece of eyewitness evidence that directly linked Harris to the shooting. Without Ward's testimony, the prosecution's case was circumstantial: None of the other witnesses could identify the gunman or place Harris at the scene, and there was no forensic or physical evidence connecting Harris to the crime. Cf. Strickler, 527 *1034 U.S. at 293, 119 S.Ct. 1936 (noting that failure to disclose impeachment material was not prejudicial in part because "there was considerable forensic and other physical evidence linking petitioner to the crime"). Appreciating the importance of Ward's eyewitness account, the prosecution featured it in closing arguments. See Strickler, 527 U.S. at 290, 119 S.Ct. 1936.
In view of the State's failure to produce the three statements, Harris's counsel had nothing to work with in challenging Ward's credibility. There was no publicly filed immunity deal; Ward's testimony seemed credible on its face as he had implicated himself as the driver in the crime spree; and Harris's key witness could not challenge the details of Ward's testimony, except to suggest that Harris was somewhere else at the time of the crime. Even though a transcript of the hearing shows that Harris's counsel anticipated that Ward must have been promised something in exchange for his testimony, he had no way to prove it. Counsel asked Ward at least six times whether the police or prosecution had made him any promises or offered him any kind of a deal, and each time Ward disclaimed any quid pro quo for his testimony. Confronted with these denials, counsel's efforts to impeach Ward fell to identifying a minor inconsistency between Ward's two police statements: whether Ward was the driver or just a passenger in the car.
That leaves us with a murder conviction in which Ward's testimony was the centerpiece of the prosecution's case, the State submitted no other evidence directly linking Harris to the crime, Harris presented a witness who denied that he was at the scene of the crime and Harris had no basis (other than the suppressed Brady material) for meaningfully impeaching Ward's credibility. Under these circumstances, it is exceedingly plausible that the disclosure of these three promises would have cast Ward's testimony, and thus all of the prosecution's case, in a different light-so different, indeed, that it undermines our confidence in the conviction and so different that the state court of appeals' contrary conclusion represents an unreasonable application of Brady. With these promises in hand, Ward had every incentive to tailor his testimony in a way the police and prosecutors found desirable. And without knowing about these promises, the jury had no basis for questioning Ward's account of events. Considerable authority from the Supreme Court and our court indicates that a defendant suffers prejudice from the withholding of favorable impeachment evidence when the prosecution's case hinges on the testimony of one witness. See Banks v. Dretke, 540 U.S. 668, 700-01, 124 S.Ct. 1256, 157 L.Ed.2d 1166 (2004); Giglio v. United States, 405 U.S. 150, 154-55, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972); Mathis v. Berghuis, 90 Fed. Appx. 101, 107-108 (6th Cir.2004); see also Bagley, 473 U.S. at 683, 105 S.Ct. 3375; Napue v. Illinois, 360 U.S. 264, 269, 79 S.Ct. 1173, 3 L.Ed.2d 1217 (1959); Joseph v. Coyle, 469 F.3d 441, 471-72 (6th Cir. 2006).
Nor were Ward's statements too informal to "rise to the level of a promise to induce Ward's testimony," as the state court of appeals concluded. Harris, 2001 WL 1004367, at *1. Brady is not limited to formal plea bargains, immunity deals or other notarized commitments. It applies to "less formal, unwritten or tacit agreement[s]," so long as the prosecution offers the witness a benefit in exchange for his cooperation, see Bell v. Bell, 512 F.3d 223, 233 (6th Cir.2008) (en banc); accord Wisehart v. Davis, 408 F.3d 321, 323-24 (7th Cir.2005), so long in other words as the evidence is "favorable to the accused," Bagley, 473 U.S. at 678, 105 S.Ct. 3375; see also Giglio, 405 U.S. at 152-53, 92 *1035 S.Ct. 763. The three police statements satisfy that requirement.
The district court also had ample grounds for accepting Harris's argument that the police made these statements to Ward. After the trial, Ward testified at the state-court evidentiary hearing that police officers had made these promises to him, and the State did not offer then, and has not offered since, any evidence to the contrary. It makes no difference that Ward ultimately gave three different versions of events: one at Harris's preliminary examination, one in an affidavit attached to Harris's motion for a new trial and one at the post-trial-evidentiary hearing itself. That Ward's preliminary-examination testimony differed from his affidavit in one respectby disclosing the exchanges between him and the policeis of course the whole point of his Brady claim and the central premise of it. And the only difference between Ward's affidavit and his testimony at the post-conviction evidentiary hearing was trivial. In his affidavit Ward said that the prosecutor told him on the day of Harris's preliminary hearing that he would be released if he testified consistently with his second statement (and that Ward should deny that any promises were made to him), while at the post-conviction hearing he said that one of the police officers said these things. That minor difference does not mandate rejecting Ward's testimony, especially when the State to this day has failed to present any testimony or evidence rebutting Ward's version of events.
The State alternatively urges us to remand the case to the district court for an evidentiary hearing at which the State can attempt to show that the promises/statements were never made. But the State never made that request to the district court and thus has forfeited it on appeal. See Enertech Elec., Inc. v. Mahoning County Comm'rs, 85 F.3d 257, 261 (6th Cir.1996). And the State has no tenable basis for saying that it was surprised by this evidence or its relevance to the conviction. Nearly eight months before the state-court hearing, Harris filed a supplemental memorandum in support of his motion for a new trial to which he attached an affidavit by Ward claiming that the police made these statements. Having failed to challenge the factual accuracy of Ward's testimony before the state courts and the district court, the State cannot challenge it now. The time to submit evidence or seek an evidentiary hearing is before factual allegations become the basis for a decision against the State, not after. Because Harris is entitled to a new trial on his Brady claim, we need not reach the arguments raised in his cross-appeal.

IV.
For these reasons, we affirm the district court's assessment of Harris's Brady claim and remand the case to the district court to dismiss claims ten and eleven and to enter judgment granting the conditional writ in favor of Harris.
