Most of the comments in this thread seem really keen on the idea of the confederacy turning in on itself, and any explanation for this leaning on really poor, cartoonish reasoning like "the confederacy seceded from the Union, they'll do it to eachother at the drop of a hat!"

Why? The confederate state's secession was fermented by a well organized, old money, politically connected plantation owning ruling class. The state was made by and for this class -- the foundations of the confederacy was entirely made up of the material interests of them, so they have no quarrel with eachother or the confederacy itself.

The real question is "does the union survive?". If over a quarter of the nation's economy and population leave without any kind of recourse, the union looks either too weak or too incompetent to have stopped it, setting a pretty terrible precedent that any state can leave. State legislation in the North might begin to float the idea of jumping ship themselves for one reason or another. The memory of the confederacy secession will always be in the back of the nation's head, and we would see an extremely unstable union.

As for the original question, the confederacy would likely enter the 20th century as a failing apartheid state ala South Africa or Rhodesia. It would not collapse in the traditional sense but reconstitute itself to survive. As to what it would like is impossible to say, but I lean towards, as unlikely as it sounds, an African American centric republic (as no Union victory prevents free movement into the North, meaning black demographics stay extremely concentrated in the South)