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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.
by McCloskey April 06, 2011
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