The 26-year-old South Africa international joined the club in 2018 and has spent the last two-and-a-half years on loan in Belgium.
After a season with Brussels-based club Royal Union Saint-Gilloise, Tau moved to Bruges last season and helped them win the league title and played in the Champions League.
Tau linked up with Anderlecht, who are managed by former Manchester City captain Vincent Kompany, last August and has scored four goals in 15 games this season.
Albion have received a Governing Body Endorsement, the new points-based system for non-English players which came into force last week when the transition period following the UK's exit from the European Union ended, from the FA for Tau.
Head coach Graham Potter said, “We're pleased to be able to welcome Percy to the club and begin working with him. I know he's a player whose progress many of our fans have followed in recent seasons.
“We too have closely monitored his performances in Belgium in the last three years. He's been playing at a really good level, particularly with Bruges and Anderlecht where he has been involved in the Champions League, and his next step is to show he is ready to make the transition into the Premier League.
“He brings some different attributes and qualities to the group and he wants to take the next step in his career. We’re looking forward to working with him.”