In Nix v. Williams, 467 U.S. 431 (1984), the U.S. Supreme Court created an "inevitable discovery" exception to the exclusionary rule. The exclusionary rule makes most evidence gathered through violations of the Fourth Amendment to the United States Constitution, which protects against unreasonable search and seizure, inadmissible in criminal trials as "fruit of the poisonous tree". In Nix, the Court ruled that evidence that would inevitably have been discovered by law enforcement through legal means remained admissible. The inevitable discovery doctrine permits the introduction of illegally obtained evidence if the prosecutor can prove that the otherwise excluded evidence would have been lawfully discovered in the course of a routine, predictable investigation.
Furthermore, under the "independent source" doctrine, the Fourth Amendment does not require the suppression of evidence initially discovered during police officers' illegal entry of private premises if that evidence is also discovered during a later search pursuant to a valid warrant that is wholly independent of the initial illegal entry. Murray v. United States, 487 U.S. 533 (1988).
Whether either of these ground could be established in this case is a tricky question. Fine details of the facts would matter as would the exact account of what the police did next.
An inevitable discovery argument might be stronger than an independent source argument in this case, where there was really no basis upon which a warrant could have been applied for and nothing that would have tipped off the police other than the discovery that they made during the illegal search that a crime was committed.
Another conceivable exception has already been ruled out. In James v. Illinois, 493 U.S. 307 (1990), the United States Supreme Court prohibited the admission of evidence obtained in violation of the Fourth Amendment for the use of impeaching statements made by a defense witness, such as the defendant asserting his innocence in testimony at trial, for example, by presenting testimony in support of a false alibi defense.
However, while James v. Illinois is a binding U.S. Supreme Court precedent, I wouldn't be too surprised if this 5-4 decision made by a much more liberal court might be overruled by the current U.S. Supreme Court which is the most conservative U.S. Supreme Court that the U.S. has seen in six decades or more, particularly in a case that provided any basis for distinguishing that 1990 decision, even if the distinction was rather dubious.
Also, this illegally obtained evidence might be available to press professional ethics charges against an attorney who knowing that this inadmissible impeachment evidence existed, allowed his or her client to testify falsely under oath in the criminal trial of his client. See Model Rule of Professional Conduct 3.3, since the search did not violate the lawyer's Fourth Amendment rights, only the Fourth Amendment rights of his or her client. So the exclusionary rule might not apply to that ethics sanction proceeding.
As a result, the criminal defense attorney for the defendant might be compelled as a practical matter to have his or her client not testify and to hold the government to its obligation to provide the case beyond a reasonable doubt with admissible evidence.
Of course, the U.S. Supreme Court is not the only source of case law related to the 4th Amendment. Roughly half of all appellate court cases in the United States are criminal appeals. One of the most common grounds for an appeal in a criminal case is that evidence should have been suppressed pursuant to the 4th Amendment but was admitted in a manner that was prejudicial to the defendant.
There are probably more than a thousand published appellate court precedents in the United States addressing exceptions to the 4th Amendment exclusionary rule that are potentially relevant to this fact pattern. There are probably at least a dozen or two cases that involve almost identical facts nationwide. Those cases would be the focus of any briefing on this legal issue in a real case. If there were a case in the state or judicial circuit where the case was going to be tried that involved almost identical facts, that case would control the outcome, at least below the U.S. Supreme Court level.