Thanks to Motor Voter, foreign nationals had to affirmatively check "I am not a citizen" when getting their US driver's license. My German room-mate did not, so he got registered to vote.
I know this because one day while getting them mail, I pulled his registration 'this is where you go to vote' card out of the mail.
The ID he used to get his driver's license was his German passport, so the DMV actively ignored the proof he provided that he WAS DEFINITELY NOT a US citizen when applying for his driver's license and registered him to vote anyway, as the law instructed them to do.
Once he was registered to vote, even an ID requirement wouldn't stop him, because the same people who gave him the ID are the once who registered him to vote. He had a driver's license and he was on the voter rolls illegally but listed as a legal voter.
I hope that the rolls are scrubbed regularly to remove people in his situation, but I don't trust that they are.