Linda Winer Newsday theater critic and arts columnist Linda Winer.

Winer is chief theater critic and arts columnist for Newsday, which she joined in 1987.

Brits on Broadway are hardly news. Brits at BAM have been a staple for what seems like forever, returning in March with the Royal Shakespeare Company in a cycle of Shakespeare’s kingly masterworks.

What we have right now, however, are Brits in Brooklyn — big ones in small, shiny places with idiosyncratic, intriguing individual projects.

On Thursday, director Trevor Nunn, who swept Tony Awards in the ’80s for “Cats,” “Les Miserables” and “The Adventures of Nicholas Nickleby,” opens Shakespeare’s rarely seen “Pericles” far Off-Broadway at the Theatre for a New Audience’s airy, modern 299-seat home at 262 Ashland Place. Despite Nunn’s vast background as artistic director of both the RSC and the National Theatre, he has never before staged this late play. He also has never before directed Shakespeare with a U.S. company, which, not incidentally, includes Christian Camargo and Will Swenson.

And Sunday, Mark Rylance — who won acting Tonys for three of his Broadway performances in the last eight years and is up now for a supporting actor Oscar for his performance as a Soviet agent in “Bridge of Spies” — will open at St. Ann’s Warehouse, newly ensconced in a historic waterfront tobacco warehouse at 45 Water St. in DUMBO.

Rylance is starring in “Nice Fish,” which he conceived, wrote and adapted with Minnesota poet-costar Louis Jenkins. This is the English actor’s first attempt at playwriting. He based it on a book of Jenkins’ prose poems about two men ice-fishing on a frozen Minnesota lake. The work had a previous incarnation in 2013 at the Guthrie Theatre, but this one, which played first at Harvard’s American Repertory Theatre and was directed by Rylance’s wife, Claire van Kampen, is theoretically the finished product.

Anyone who knows the actor’s restless, unconventional brilliance will understand why I say theoretically. You may remember his first Tony speech, for his 2008 Broadway debut in “Boeing, Boeing.” That was when he baffled the national audience by not thanking his teachers and his agent. Instead, he said “When you’re in town, wearing some kind of uniform is helpful . . . at the very least you should wear a suit and carry a briefcase and a cellphone . . . ” and continued like that for several other befuddling moments.

We learn now that he was reciting a poem by Jenkins, whose wild, wilderness sensibility he had admired for years. He read a different Jenkins piece when he picked up his 2011 Tony for “Jerusalem,” but — don’t get too comfortable with expectations — gave a touching, comprehensible speech when he won for the dazzling all-male “Twelfth Night” in 2014.

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And now, ice fishing in the Midwest? You see, Rylance, 55, was born in England and spent most of his career there — including his influential decade, beginning in 1995, as artistic director of Shakespeare’s Globe.

But he also lived from age 9 through high school in the unlikely suburbs of Milwaukee, and he fondly remembers playing ice hockey on a frozen lake. The experience of “Nice Fish” involves what he has called “a bonding place for men out on the ice. I was interested in why men go out there.”

In a written statement, he explains more. “Having come to Broadway five times with English productions, I’m thrilled to be here this time with a brand-new American play, my first original play in America. . . . The humor, insight and eloquence of the long Midwest winter is something I remember and love from my teenage [years] in Wisconsin. This play could only come from America.”

There is also an American connection for Nunn and “Pericles,” which has an original score by Tony-nominated composer Shaun Davey (“James Joyce’s The Dead”) and a 22-member cast that includes the PigPen Theatre Co., downtown storyteller-musicians. According to the TFANA website, Nunn has wanted to do this adventure-packed Shakespeare in New York for years. “I wanted to do something here that is relatively unknown and not at all about English history — a play where there’s absolute permission to be fictional.”

Nunn, 76, was virtually an annual Broadway presence with musicals and serious plays for three decades. But his last Broadway production was the revival of “A Little Night Music” in 2009. Except for an exquisite Beckett miniature, “After the Fall” (with Michael Gambon and Eileen Atkins) for a brief run far Off-Broadway in 2013, he has done nothing in this country.

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Jeffrey Horowitz, founding artistic director of TFANA, tells me that Nunn’s “Taming of the Shrew” was the first live Shakespeare he ever experienced. “It was funny, sexy, fresh, human. It inspired me to study acting in London,” which led to him starting this theater. “Now Trevor is here in our new theater staging a play about a journey. That feels right as he was a beacon at the beginning of mine.”

There are very different lives after Brooklyn for both Nunn and Rylance this summer. In May, Nunn — who, not T.S. Eliot, actually wrote the lyrics to “Memory” — begins rehearsals for the Broadway return of “Cats.” Rylance stars as the Big Friendly Giant himself in Steven Spielberg’s adaptation of Roald Dahl’s “The BFG,” set for a July release. And there will be no performance of “Nice Fish” on Oscar night, Feb. 28. Wonder what he’ll say if he wins.