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With a PoD no earlier than 1901, have a man so nominated by the Socialist Party (without giving him the concurrent nomination of the main Democratic or Republican parties) win the Presidency of the US -- by 1920, but preferably in 1912.
Note, this man does not have to be Eugene Debs -- in fact, I think I think it would be more plausible if it wasn't. Also, bonus points for making him a two-termer.