re: Constitutionally speaking, what is the correct way to handle a coup?

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During the Civil War the rebels didn't attempt to usurp the federal government, they instead declared their own separate gov, so it wasn't exactly a coup per say.

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There is a term for that: succession. They broke away and claimed to have done it legally.

This is why a lot of people call Lincoln a tyrant. The argument is that the southern states had a legal right to break away and the remaining United States were the aggressors by going to war to bring the south back into the Union.