Some call Mark Rylance mad. He himself is happy to be called eccentric, for he’s set out to experiment, surprise and challenge convention.

Rylance is starting his last summer as artistic director of Shakespeare’s Globe with a season called The World and the Underworld with a revival of The Tempest in which period singers and modern dancers outnumber the performers, of whom there are just three.

He plays Prospero, Alonso, Sebastian and the drunken butler Stephano in a production packed with mythic overtones, Jungian undertones — and, since he can be as funny onstage as he’s earnest offstage, plenty of wry humour.

For me, Rylance is a genius. Since he took on the job in 1996, like a scientist in a lab, he has tested a deeply unfamiliar theatre’s inner workings. It would have been easy to opt for that dismal thing, “a safe pair of hands”. That way, the replica of Shakespeare’s most famous playhouse would have become a heritage site, a Bardic theme-park and tourist trap. No chance of that with Rylance.

Yet challenging convention hasn’t meant rejecting tradition. On the contrary, Rylance’s tenure has been a serious attempt to discover what Elizabethan staging was like and what it can teach us now. Ask Rylance what he’s learnt, and he’ll tell you that Globe performers must be vocally unfussy, as fleet and sharp with words as footballers passing the ball, physically as nimble as dancers, always aware that they’re telling bold, tense stories, and vital enough to touch the audience’s hearts, not just its heads, with “the visceral joy of Shakespeare”.

“But the main thing I’ve learnt,” he says, “is to think of the audience as another actor, reacting to it, sharing scenes with it, absorbing whatever it does into the reality of the play. That’s easy to do with the courtroom scene in The Merchant of Venice or the army in Henry V. But even when you’re on your own, they can be the face of one’s conscience, watching the story of one’s life. You are saying to them: “To be or not to be, that is the question, isn’t it?” So now I always say to actors: don’t speak to the audience or for the audience. Speak with the audience.”

Since some spectators are only inches away, this demands both courage (“I’ve seen actors in absolute terror before going on”) and flexibility. Performers must adjust the decibels of a love scene when planes pass overhead and respond instantaneously to groundlings who vary greatly from performance to performance. The old problem, that of boorish, distracting cries of “f*** the French” and so on, rarely surfaces now, for familiarity with the Globe has matured audiences as well as actors. But Rylance, while somehow projecting his voice to the top of the theatre’s three-tier cylinder, has found himself making eye-contact with weeping oldsters and laughing youngsters — and nodding gravely at the boy who murmured “yes” when his Hamlet asked “Am I a coward?”

This is obviously logical in a theatre where everyone shares the same light but, for Rylance, it’s also an artistic plus. He doesn’t think he can again play Shakespeare in what he calls a “closed-box theatre”, where actors aren’t only disconnected from the audience but, thanks to spotlights, often can’t see them. What he’ll do after this farewell season — The Tempest, Pericles, The Winter’s Tale and Peter Oswald’s Plautus-derived comedy The Storm — he doesn’t yet know. But he talks of taking jointly created new work to unconventional spaces.

Indeed, when you ask him what he’s achieved at the Globe, his first answer is “create a community of collaborators”. He’s formed a team that includes figures like “the master of voice”, meaning someone whose relationship with the director or “master of play” resembles that of an opera conductor with his producer, and he’s ensured that musicians are as important as they surely were in Shakespeare’s day. It’s a policy that has come close to packing out the theatre — it played to 93 per cent of capacity last year — and has brought the Globe productions Rylance can remember with pride: Antony and Cleopatra to Twelfth Night, King Lear to “the thing I’m most delighted with”, Oswald’s adaptation of Apuleius’s Golden Ass.

Myself, I’ve admired revivals that conform with “original practices”, meaning that costumes, props, everything, are as authentically Elizabethan as possible, and “modern practice” productions that give directors free range in an antique space. The one has given us Rylance’s own Cleopatra and award-winning Olivia, the other Vanessa Redgrave’s Prospero, among other treats. Myself, I’ve mixed feelings about his occasional all-female casts, since they have been uneven, but they’re a good example of his creative daring.

They also allowed Globe audiences to do what they seem to find most exhilarating: stretch their inner muscles in a theatre that, under Rylance, has become a gymnasium for the imagination. The only downside is that some of our finer directors and classic actors still won’t work there. Shame on them, say I. If Rylance’s reign has proved anything, it’s that testing one’s skills in Shakespeare’s own crucible should be as professionally essential as an Equity card.

The Tempest is at Shakespeare’s Globe, London SE1 (020-7401 9919)