The alternate history challenge, should you choose to accept it, is to have the US become a North Korean style dictatorship.
Well, since this is after 1900, here's a hypothetical scenario: there's a more successful Klan and/or pro-fascist(think: Charles Coughlin, Gerald L.K. Smith, William Pelley) during the interwar years, first off. Then, perhaps the Allies lose World War II, and some President(not FDR) blames the USSR for the loss, and a Cold War more heated than IOTL begins to take shape.
Nazi Germany implodes some 15 years later thanks to near constant revolts, economic collapse, and just plain incompetence. Both the Soviets and the Americans rush in to try to take advantage of the collapse, and, eventually, at some point in that same decade, the 3rd World War breaks out over some crisis, which may directly or indirectly related to the collapse of Germany. Nukes fly, and both countries are devastated. The U.S. government is decapitated, and slowly, over the next decade, the country collapses(as does Soviet Russia) until the country finally just stops existing in it's old form as the Seventies come to a close.
Several new nations are formed, but the rump government claiming to be the true successor to the old U.S., re-coalesces, in the South, of all places, or at least portions of it(for the purposes of this scenario, though, really only southern Ark., northern La., Ala., Miss., Ga., far northern Fla., S.C., central and eastern N.C., and parts of Va. adjacent to N.C.). Tragically, it ends up turning into a complete hellhole, as it is not just basically run by a revived Klan and White Citizens' Council, but has also attracted hardcore racist & other reactionary elements from elsewhere in the former U.S.A.; many more moderate and liberal whites either flee or are forced into labor camps, and the state attempts to indoctrinate their children. Minorities have it even worse, as many are shipped off to farms to work almost to death for virtually nothing. And the military becomes ubiquitous; in many places, they are judge, jury and executioner. And even many whites in general are dirt poor and have to work for shit wages but the government always cheers them up with racialist propaganda and hate fests a la Orwell's "1984", amongst other things.[1]
In general, this rump U.S.A. genuinely horrifies and outright disgusts the other American polities, but as with the real world North Korea, this situation could last a while, potentially until they actively provoke a shooting war with somebody.
I realize this is far from the most likely scenario, but it does match the criteria, I think.
[1]Just after I finished writing this, it struck me that I may have been subconsciously influenced, in some ways, by the story of some of the neo-Confederate enclaves that developed after the bombs fell in 1983: Doomsday.
Edit: here's a link-
http://althistory.wikia.com/wiki/Taccoa,_CSA_(1983:_Doomsday)