Slang meaning, "converted to Roman Catholicism." The Tiber being a river that runs through Rome and thus "swimming" over to the Vatican side.
Guy#1: I've never seen you before, Are you new to this Church?
Guy#2: Yes, I've just recently swum the Tiber.
Guy#1: Oh, congrats, what were you before?
Guy#2: Anglican.
Guy#2: Yes, I've just recently swum the Tiber.
Guy#1: Oh, congrats, what were you before?
Guy#2: Anglican.
by McCloskey April 7, 2011