Julie Taymor has settled on “A Midsummer Night’s Dream” as the play she’ll helm to inaugurate the new Brooklyn homebase of Theater for a New Audience this fall.The Off Broadway production “Midsummer” will be Taymor’s first gig for the Gotham stage since her high-profile stint as helmer of Broadway’s “Spider-Man: Turn Off the Dark,” followed by her subsequent messy divorce from that project. For several months Taymor has been on tap to launch the soon-to-be-completed TFANA building in Fort Greene, Brooklyn, but it hadn’t yet been decided which Bard play she’d choose to direct. Taymor (“The Lion King”) has long been associated with TFANA, beginning with a 1986 staging of “The Tempest” and followed by “The Taming of the Shrew,” “Titus Andronicus” and “The Green Bird,” a 1996 staging of a Carlo Gozzi play that moved to Broadway in 2000. TFANA’s new building — the first permanent home for the troupe, which was founded in 1979 — will include a 299-seat mainstage and a 50-seat second space. Casting and other details for “Midsummer” have yet to set, but previews begin Oct. 19 ahead of a Nov. 2 opening.