The men were asked to leave the City Arms and after originally thinking it was a joke, the priests attempted to prove their profession before making for the exit.
As the priests left, they were approached by a member of staff who realised they were telling the truth. His apology was greeted by a round of applause from locals.
Speaking to WalesOnline, Matt Morgan, assistant manager at the City Arms, said: “When we realised our mistake we quickly apologised to the priests and thankfully they were all great sports and saw the funny side. “We offered them a free round of drinks and they were happy to stay. “Our Rev James ale is one of our most popular beers so it was great to have a real-life Reverend James and his fellow priests enjoying a pint or two!”