Syd Pollock bought the Havana Red Sox from Ramiro Ramirez in 1928. Ramirez stayed on as the manager and the team began barnstorming around Miami. In 1930, Pollock changed the name of the team to the Florida Cuban Giants and soon after Pollock introduced comic routines into the games and developed what was to become known as "shadow ball." Shadow ball was when the infielders would mime throwing a ball around for between-inning warm-ups. These routines would later be made famous in the 1940s by Pollock's Indianapolis Clowns and Abe Saperstein's Harlem Globetrotters basketball team.
In 1931, they were an associate team in the Negro National League:5 and operated under the Cuban House of David name which Pollock appropriated from the original House of David, a white commune known for their bearded baseball players. The team joined the East–West League in 1932 as Pollock's Cuban Stars. The returned as an independent team still under the "Pollock's Cuban Stars" moniker from 1933 until 1936.