I suspect that in itself may be significant, in that it suggests that the comedies are the most often played and initially resorted to by someone who's familiar with the space...

What I meant to say was that "Much Ado About Nothing" is the only one of the shows I can think of that I've seen at the Globe that did have any tragic content -- the others have been pretty much straight comedies. But then the tragic element did work very well. (They had Hero's 'tomb' at the end of the thrust stage extended into the audience, and a long silent sequence when the mourners processed through the audience to make penance there -- I was literally close enough to the actors to have touched them. They also used this space for Juana to do her plotting and have secret rendezvous.) And as you mentioned the period music can be very eerie.