What was “Sweeney Todd: A Musical Thriller” is now “Sweeney Todd: A Pie Shop Sideshow.”

Stephen Sondheim’s 1979 musical is a gory and glorious concoction of over-the-top melodrama, English music hall, horror movies, opera, black comedy, social criticism and breaking of the fourth wall.

It can be staged and interpreted in infinite ways, ranging from Hal Prince’s supersized original 1979 production set around an iron foundry to a downsized 2005 Broadway revival where the cast doubled as the orchestra.

Tooting Arts Club’s new production, which originated in London and is now playing off-Broadway in the West Village, is built around a working pie shop, not unlike the one operated by Mrs. Lovett, Sweeney’s money-hungry partner in crime. Audience members can dine on meat pies (prepared by a former White House pastry chef) before the show.

The eight-member cast combines holdovers from the London production (who are staying with the show only through April 9) and Broadway veterans. They are joined by just three musicians.

The pie shop concept is cute at first, but “Sweeney” is a heavily-plotted drama that involves many different locations. This production is awkwardly staged around just a service counter and communal tables, and few seats are not obstructed in some way.

At its best, “Sweeney” can be terrifying, mesmerizing and sweeping, but this production sacrifices the storytelling and score for the sake of a gimmick.

The cast is decent but not noteworthy. Jeremy Secomb is a restrained Sweeney who relies mainly on a quiet and grim exterior, while Siobhán McCarthy makes for a hyperactive Mrs. Lovett. Beginning April 11, they will be replaced by Norm Lewis and Carolee Carmello.

Perhaps someone can find a way to produce “Sweeney” in the elaborate and immersive style of “Sleep No More,” bringing audience members not just to the pie shop but also the barber parlor, bake house, mad house, docks and crowded streets of London.