When you’re Taylor Swift, keeping a new romance completely under wraps is no easy feat.
“She wanted to get to know him without any chaos,” a source tells PEOPLE exclusively about Swift, who was longtime friends with the rising star before things turned romantic.
According to The Sun, Swift has been renting a house in North London and donned disguises while spending time in the U.K. with Alwyn, who’s been busy shooting his upcoming film, The Favourite, with Swift’s friend Emma Stone.
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Last year, Alwyn — who had his breakthrough role in Billy Lynn’s Long Halftime Walk (Swift caught a screening of the film last November with her mom Andrea and the Haim sisters) — opened up to PEOPLE about his growing fame.
“It’s really nice to have people excited about … me,” he said about the newfound attention. “But it’s hard to get caught up in that. Maybe it’s just my nature. I just want to take things a day at a time.”