Shea received a work permit after a hearing in London and Stoke manager Tony Pulis is delighted to have secured his signature.
"It's fantastic news that we have been able to push through the deal," Pulis told the club's official website. "Brek is a very, very exciting player, and one who we firmly believe could go on to become an outstanding player for this Football Club.
"He has terrific pace, a very good left foot and is capable of scoring goals. He has all the tools needed to be a success here."