Executive branch refuses to comply with judicial decisions

The underlying case is about immigration. An undocumented immigrant, Jorge Baez-Sanchez, was subject to removal, but federal courts decided that he was eligible for a special visa and could remain in the United States. The executive branch declared the decision invalid, and refused to comply.

While the particular case is not too significant, the precedent is potentially huge. If the executive branch just decides not to comply with a judge's decision, what happens next? There is not much you can really do as a judge. They have access to the US Marshals and can attempt to arrest the executive branch members, but US Marshals are part of the executive, so it's unclear what orders they would follow. Congress in theory could act, but that seems highly unlikely (see impeachment)

The official decision is here: http://media.ca7.uscourts.gov/cgi-bin/rssExec.pl?Submit=Display&Path=Y2020/D01-23/C:19-1642:J:Easterbrook:aut:T:fnOp:N:2462983:S:0


JUDGE EASTERBROOK: What happened next beggars belief. The Board of Immigration Appeals wrote, on the basis of a footnote in a letter the Attorney General issued after our opinion, that our decision is incorrect. [...] the Board flatly refused to implement our decision. Baez Sanchez has filed a second petition for review. We have never before encountered defiance of a remand order, and we hope never to see it again.

Supposedly the judge in this case is a conservative judge.