POD: Instead of openly proclaiming Cuba communist and seeking the shelter of the Soviet Union, Castro uses the Catholic religion to temper the new socialism and expose Batista's regime as an immoral dictatorship. By inviting the US in to view the evidence rather than start killing off the Batista-ists outright, a two-party government (Castro's "Catholic Socialists" and the US backed "Democratic Cuba") is recognized by the US and gives internationally observed trials for the lot. Republicans accuse Kennedy of being soft on Communism when Catholicism comes into play, but a precedent is set and the spread of Catholic Socialism through Latin America (Che unleashed) is frowned upon by the US but seen as a buffer and preventive of the Soviet Union spreading influence in the Americas. Castro's party eventually becomes the minority one but remains influential in government and pervasive throughout Cuban culture and day to day goings on.
The Mexican Catholic Socialist party becomes an influential minority party in Mexican politics as well.