At a backstage tour at the Oregon Shakespeare Festival, my wife and I once got to see an interesting green room that served two different theaters at the site. We heard how that week you might find actors costumed for Titus Andronicus hanging out between scenes with actors ready for Saturday, Sunday, Monday, a play set in modern Italy.
The idea of these two very different sets of Italians mingling in the same green room seemed like a wonderful setting for a story or play. Or, pick a different pair of productions to play off of. It's a kind of game -- take a Shakespeare title and match it with a play that makes a contrast, and mix the actors together with behind-the-scenes issues, something like Noises Off. Hamlet meets Endgame, King Lear meets Death of a Salesman, Henry V meets Pygmalion. And of course someone should get so worked up they stomp out onto the wrong stage.