In the rehearsal room of the Irishtown Players, the posters on the walls are a sampler of the company’s performance history: 20th-century classics, almost all. Brian Friel’s “Dancing at Lughnasa” is up there, of course, and Samuel Beckett’s “Waiting for Godot.” Martin McDonagh’s “The Beauty Queen of Leenane,” too, and Conor McPherson’s “The Weir.”

The only unfamiliar title, “The Happy Leper of Larne,” almost winks from its frame, suggesting a maudlin-cheery cousin to “The Cripple of Inishmaan.”

For the Irishtown Players — the fictional Dublin troupe at the center of Ciara Elizabeth Smyth’s new backstage comedy, “Irishtown” — “The Happy Leper” was a hit. Now some producers are bringing the company to Broadway in the author’s follow-up play. But with mere weeks until they leave for New York, the playwright, Aisling (Brenda Meaney), has gone rogue. Her just-delivered script is a contemporary legal drama about sexual assault, set in England.

To Constance (a flawlessly funny Kate Burton), the ranking company member, such a play is not Irish at all.

Poppy (Angela Reed), the play’s British director, points out that by definition it is, because Aisling is.

“Yes,” Constance allows, her voice rising theatrically, “but where are the rolling hills, where is the bar, why is everyone alive?”