Every two years, the federal government certifies that the 50 states have an acceptable form of government. This sounds odder than it is. Article 4 Section 4 obliges the federal government to guarantee "to every State in this Union a Republican Form of Government". Luther v Borden, a case over the Dorr rebellion in Rhode Island established that this guarantee is non-justiciable, that the Supreme Court cannot intervene.
This guarantee is what was used to overthrow the elected governments of the Confederate states after the civil war and put the Army in charge under Reconstruction. With the passage of the 14th amendment, the equal protection clause of that amendment became the dominant way for the federal government to control states who strayed too far from american ideals. Yet this dependence on the 14th amendment is convenience, not any sort of overthrow of the clause.
These days the interesting theoretical question about this clause is whether there is any sort of government that does not violate equal protection but does constitute a violation of the republican form of government guarantee. It seems clear that it is. The establishment of a system that made the votes of the people essentially worthless would seem such a trigger but only if the situation became intolerable enough to provoke rebellion.
The interesting practical question is whether there is a threat that could actually trigger it. The most likely big issue seems to me to be government pensions swallowing up governments. Past a certain point, government ceases to be anything meaningful beyond cutting transfer payment checks. The will of the people is functionally nullified. For this to actually become an issue, only the Congress could step in either by simply not recognizing the validity of a state's federal representatives or by more involved legislation like the reconstruction act of 1867.