They need a warrant to search unless someone gave consent, there are exigent / emergency circumstances or it is incident to arrest. Does not sound particularly legit to me. Your friend needs an attorney.
Consent may be implied if your friend was on parole or probation and had given a search waiver .
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Whether the search was legal is going to depend on whether or not the police had a warrant (or a provable valid reason to bypass a warrant such as exigent circumstances) and if they did have a warrant did they stay within the bounds of the warrant. Unfortunately, whether or not the police stated aloud that they had a warrant is not really the issue.
When a person is arrested away from a location and the police have reasonable cause to believe that a person at the location will destroy evidence prior to the obtaining of a warrant, they can legally "freeze" the location pending the issuance of a warrant. That's how they had the wife wait outside while they got the warrant. They can enter and do a "protective sweep" of the location to make sure there's nobody inside before they then lock down the house and await the warrant. It's not a full search authorization though - it just allows them to do a search of the location for any people inside.
There is no legal requirement they actually show the warrant, but it had better have existed at the time of the search in this case. If they did a full search without it under these facts, the defense would have an excellent shot at suppressing any evidence found in the house.
They are supposed to leave a detailed list of what was seized, but they may have skirted the edge of the requirements by putting a copy in the arrestee's property at the time of booking, etc.
With every search question, the specific facts will matter. The defense attorney is going to have to examine the warrant and all the other discovery to fully assess this case.