There are worrying signs: the hawkers have display baskets that contain mini-champagne bottles (pounds 6) and salmon bagels (pounds 3). Before the play, actors in masks mingle with the audience, doing silly bits of interactive business. And over-anxious grey-haired ushers tell you there's only five minutes till the play starts, please don't walk that way, and no, could you not sit on the stairs. This could be the National Trust.
It makes you wonder if the Globe realises just how important a role it has to play, not within the heritage industry, but within British theatre. Thankfully, there is no chance of recreating the Elizabethan theatre-going experience: daft ambition, that it is. Slipping past the TV presenter to take my place with the groundlings (pounds 5), I didn't join a heaving, sweating mass of drunk, illiterate, smelly 16th-century Londoners. I joined half a dozen other innocents who hadn't heard the weather forecast.
It is not an Elizabethan-themed experience that the Globe ought to offer, but something something decisively new. When Mark Rylance steps forward to deliver the prologue of Henry V ("O for a Muse of fire ... " etc.) there is a thrilling appropriateness. The lines work better here than anywhere else. This is the setting they were written for. Rylance says (in effect), "Listen, there's just us up here, with no tricks and only a couple of marbled pillars to hide behind, and we want to take you to France and back. All you have to do is pay attention."
There are ravishing costumes (from designer Jenny Tiramani), three entrances, a trap-door and a balcony. That's all. In the way it restricts the action, bringing us close to the dramatic circumstances for which it was written, the Globe rejects the director-designer axis that has dominated for so long and returns us to the actor's words playing on an audience's imagination.
With Rylance's Henry, this is exactly what happens. He is a superb Shakespearean, with an engagingly hesitant manner that invites us into his mind. He suggests again and again that he has reached a crossroads, and that his next thought could go either way. In this lovely, intimate theatre, he finds a stillness and poise.
If the audience is expected to listen hard, then the actors are expected to speak well. The most important project the Globe should undertake - leaving aside the exhibition hall, research centre, educational programmes, cafe, etc - is to establish a crack team of Shakespeare actors who can handle the complexities of his verse with the skill and expressiveness of concert soloists. If they had that, then this project - hearing these plays in this particular space - would make glorious sense.
The Henry cast, in this respect, are a good deal better than Winter's Tale cast, with strong performances from a begrimed John McEnery as Pistol, and a proud, hirsute David Fielder as Fluellen. The boys play the girls in this all-male production, and Toby Cockerell as Katherine and Ben Walden as the gentlewoman Alice are funny. Richard Olivier directs a solid, pageant- like production that inspires appreciative cheers and boos from the audience.
The Winter's Tale, directed by David Freeman, has a burnished North African feel, with red earth and modern tractor tyres for Leontes's throne. While it is a mistake to make the costumes plainer than the back wall, Freeman stages the sheep-shearing festival enchantingly, with dozens of sheepskins covering the earth. As the itinerant thief Autolycus, the excellent Nicholas le Prevost sings his songs as a strained Bob Dylan balladeer. But too often, elsewhere, the verse is forced for less obviously comic reasons.
The Fifties musical Damn Yankees, by Adler and Ross, takes the Faust legend and turns it into one about a baseball fan who sells his soul to the devil in return for playing a starring role in the Washington Senators' victory over the Yankees. Like one of those old-style celebrity vehicles, this Broadway revival is tailored to the star presence of Jerry Lewis as Applegate, or the devil. Now aged 71, Lewis plays off his fame, breaking into his nutty-professor voice, to the delight of the audience, and wandering round with the predatory silkiness of a Savile Row tailor. With each exit he gave a little skip and a tilt of the head.
In the second act Lewis has his own number ("Those Were the Good Old Days") when he hijacks the show for quarter of an hour, tossing canes in the air, failing to catch them and then - as if to cover up the embarrassment - telling joke after joke after joke (some funny, some old). Vaudeville returns with a vengeance. Damn Yankees is a very enjoyable, hummable show.
The Soho Theatre not only stages new plays, it also offers - as it elegantly terms it - "a comprehensive writers' workshop programme" and "script surgeries", which give "one to one dramaturgical support". Skeleton, by Tanika Gupta, was "inspired" by a story by Rabindranath Tagore, about a medical student returning home to his Bengali village, and taking possession of a skeleton that comes to life with a past of its own. I was intrigued watching this undramatic and over-explained piece by what people talk about during "one to one dramaturgical support".
'Henry V' & 'The Winter's Tale': Shakespeare's Globe, SE1 (0171 344 4444), in rep to 21 Sept. 'Damn Yankees': Adelphi, WC2 (0171 344 0055). 'Skeleton': Soho Theatre, W1 (0171 402 0022), to 21 Jun.