Answer: They don’t have their own army and rely on good faith, tradition, fear and the Executive Branch of the government to make sure their decisions are followed. This is certainly the case in the United States.
Usually, this question doesn’t arise. When it does, it often brings up the concept of a “constitutional crisis”.
At least in recent (well, relatively recent) memory, this has happened only in the late 1950s and early 1960s. When the Supreme Court decided in Brown v. Board that schools should be integrated and a number of southern governors stated that they were going to ignore the decision. When it became clear that the governor’s were serious, the federal government did what it had to do to enforce the order of the court. It didn’t send in the clowns, it sent in the troops to order the clowns to enforce the decision of the Court. (Whether, it was ultimately successful is for another day.) And their task was relatively easy, so to speak. “Let these kids walk through that door.”
Now, we might be faced with another crisis. The states whose voting practices had been subject to the broad Voting Rights Act are now, since the Republicans and the Court decimated the Act, out of compliance practice. They thought the Act, like Reconstruction, was an artifact of the past.
But lo and behold, when the latest Alabama redistricting map was challenged, and when that challenge rose to the Supreme Court, the Court surprised everyone by ruling 5-4 in favor of the challengers. The Court decided that Alabama must redraw its map to provide for a second black majority district.
Like the southern governors of 75 years ago, the Alabama legislature refused to comply.
Now, what is going to happen? You can’t move the military in and order the legislature to redraw its map, can you?
Are their other remedies open to the federal government? If so, what? Lock out the legislature? Refuse to seat the AL representatives in Congress? Tell the people of Alabama they can’t vote? Take away federal benefits? And how would these be received? And will the Biden administration do anything and, if they do (or if they don’t), what will be the political reaction?
I haven’t yet seen it discussed in this manner, but……do we have a Constitutional crisis?