The Supreme Court has said that you can never suppress the body of a person accused of a crime – the person’s identity is not able to be kept out of evidence, even if that identity is the result of an unlawful arrest or search.
This is a huge issue in illegal reentry cases. If a person is deported then returns to this crime, that’s illegal reentry. If the person is deported after having been convicted of certain kinds of felonies – whoa buddy, that’s illegal reentry after having been convicted of an aggravated felony.
In light of the Supreme Court’s rule about how you can’t suppress the body of the person accused, many people who handle illegal reentry cases find them massively depressing. If you can’t suppress the person’s identity, even if the knowledge comes from an unlawful search, then you’ve gutted the Fourth Amendment for people accused of illegal reentry.