“Bergman Island” is the title of a new film from the writer and director Mia Hansen-Løve (“Eden,” “Things to Come”). The movie concerns a filmmaker couple, Chris (Vicky Krieps) and Tony (Tim Roth), who travel to Fårö, the island in Sweden where Bergman lived and worked, to reside for a spell in Bergman’s house, which he had wanted to be kept as a place where artists and scholars could stay. (Interested in applying? Go here!)

When they arrive, they’re told it’s good that they found the place, because no one around would have given them directions—that’s part of a tacit pact among the locals about all things Bergman. But from what we see, practically everyone on the island can’t wait to share trivia with Chris and Tony. Did you know that Bergman had nine children by several different women? That he had directed 25 films by the age of 42?* That the couple will be sleeping in the bed from “Scenes From a Marriage”? That Bergman didn’t really consider “Through a Glass Darkly,” “Winter Light,” and “The Silence” a trilogy?

As Chris and Tony prepare to watch a Bergman movie in the filmmaker’s private screening room, they argue over which one to choose, rattling off title after title, apparently for the benefit of a hypothetical viewer who has wandered into a movie called “Bergman Island” with no idea of who Ingmar Bergman is. Because none of this Bergman info is especially insightful or even obscure, watching the first section of “Bergman Island” is like being marooned with a group of talkative, know-it-all video store clerks. Fortunately, the island scenery—which is lovely, thanks to the wonders that the cinematographer Denis Lenoir works with natural light—gets Chris’ creative juices flowing, and her screenplay becomes a movie within the movie. Her script concerns yet another film director, Amy, who is attending a wedding on Fårö with an ex-boyfriend, Joseph she still loves.

This Amy-Joseph plot is instantly more engaging than everything with Chris and Tony, even if the story Chris is writing veers closer to rom-com (or at least rom-drama) than anything Bergman ever made. And mercifully, when these characters go skinny-dipping, running nude from the camera in imitation of Harriet Andersson in Bergman’s “Monika,” nobody spells out the connection in dialogue.

“Bergman Island” increasingly plays up its metafictional elements, and it’s interesting to ponder whether the romantic tension between Amy and Joseph would be as effective if it weren’t surrounded by a completely inert framing narrative. In any case, there’s a good film inside “Bergman Island”.