Based on a novella and short stories by Hanif Kureishi, Intimacy features Rylance as a barman and Fox as an unfaithful wife who indulge in bouts of loveless sex in his dingy lodgings. For the first time in mainstream British cinema, the film shows an erect penis and — briefly — oral sex.

When I meet him, Rylance is affable but very slightly cagey. Articles by Alexander Linklater, the journalist partner of Kerry Fox, have just hit the newsstands. These unpick in minute detail exactly how it feels to watch your lover apparently having intercourse with someone else — and in public. Rylance won’t say what his wife, the composer Claire van Kampen, thinks about it all, but it must have been a joint decision to accept the invitation from director Patrice Chereau to play such emotionally bruising roles.

There are tales of a dinner party during which the couples discussed the implications, public and personal, of taking part in so explicit a movie. Surely they must have expected particular scenes, especially the one in which Fox’s character, Claire, takes Rylance’s penis in her mouth, to stir up prurient media interest? Rylance has a boyish face belying his 41 years and a way of arching his eyebrows which suggests an almost other-worldly innocence. “No, I didn’t know they’d focus on that rather than other aspects — perhaps naively,” he says. Trusting he may be, but he is no fool. He doesn’t do anything without serious consideration. “It will be a shame if the form rather than the content is what people talk about. It’s so easy to spin a story from it.”

For Rylance, the “content” means a serious discussion of the difficulties and mysteries of intimacy. The question raised is whether, in an age where sex seems all-pervasive, physical love-making between two virtual strangers, who meet regularly on Wednesday afternoons but scarcely speak to each other, can lead to emotional involvement.

“It is a thoroughly considered question in a movie which reveals the difficulty of achieving intimacy with all the inquiring detail of a French film. I hear it’s been well-received on the Continent, but perhaps they’re better coping with such things there.” It is after what Rylance refers to delicately as “the unusual act that Kerry does” that his character, Jay, realises he has never loved anyone as much before.

This is not a pornographic film, not intended to be arousing. Shot in the basement of a squalid flat where Jay lives after walking out on his wife and family, it does not attempt to glamorise either characters or location.

The sex scenes were all filmed in one intense week at Pinewood studios: “Patrice pushes you to go further emotionally so that it is only when you have gone too far that he says, fine, we’ll use the previous take.”

These sensitive scenes were filmed on a closed set with only the director and a cameraman in attendance. The camera was not hand-held so the actors had some idea of what was in frame and each of the six sex scenes was carefully choreographed, rehearsed and played in real time according to detailed instructions in the script. The actors talked, trusted and respected each other. “Kerry,” says Rylance, “is very centred. You always know where you are with her. She is not neurotic or fussy.”

Fox is reported to have said that a near-rape scene was especially difficult for Rylance. That isn’t quite how he remembers it: “That was a difficult day. It was very shocking for her. I just went quiet. But you know, I hope people will be involved in the story, not thinking how we did the film, after the lengths we went to . . . Oh dear, I’m afraid it seems to be difficult to avoid innuendo in the circumstances.

“But I think the scenes between Jay and Andy (Claire’s cuckolded husband, played by Timothy Spall) are equally appealing. They show the wrecking nature of Jay’s character. There are a lot of very primal things in this film. It is anti-pornography; it shows its limits.”

Rylance was not, he says, the first choice for Jay. However, Chereau liked his performance in Gillies McKinnon’s film about drug pushers, The Grass Arena — “and Cleopatra,” adds Rylance, with obvious pleasure, remembering his success as the cross-dressed Egyptian queen at Shakespeare’s Globe two years ago: “Patrice said he could see that I was willing to go deep, to commit emotionally.”

It is months since Intimacy was completed and Rylance is now deep in rehearsals for Mike Alfred’s six-hander Cymbeline at the Globe, in which he plays the insanely jealous hero Posthumus and his rival in love, Cloten. Everyone will take two or three parts in a storytelling production.

During his six-year tenure as artistic director at the Globe, Rylance has played Henry V and a number of minor roles as well as a memorable Hamlet and Cleopatra. Between filming Intimacy and the opening of this year’s season, he collected excellent reviews for his role in the National Theatre production of Yasmina Reza’s Life x 3. He may be reasserting himself as an actor, but the Globe job is close to his heart.

He’s a passionate defender of the place, particularly in the face of criticism from some quarters that the Globe encourages inappropriate audience reaction, reduces tragedy to pantomime and is flooded with tourists only there to see the building.

“I dislike the phoney respect for Shakespeare you find in some theatres, the audience hushed, thinking they’re in the presence of high art. Why shouldn’t the audience respond? The past 100 years of criticism have removed some of the visceral joy of Shakespeare; shepherds and fools have an important role in the plays.

“Besides, why do the critics never write about people crying or being absolutely still at the end of the play? They come on one night and we play to 250,000 people over the next four months. Every performance is different because every audience is, and that is especially so at the Globe — which is why we video every third one and people come from all over the world to study them. If visitors come initially to look at the building I don’t care — they stay to watch three hours of Shakespeare.”

Does he invite first-rate young directors? Yes, he says, and he mentions some star names, Matthew Warchus and Katie Mitchell among them, who have turned him down. He acknowledges that the Globe is a demanding space for actors and that he lays down certain ground rules for directors about the collaborative nature of working at the Globe, which might put people off. But directing Shakespeare in the intimate confines of the National’s Cottesloe or the Royal Shakespeare Company’s Swan cannot, he believes, compare with the Globe experience: “The rewards are much greater from the audience. They are so quick.”

The current, highly stylised production of Macbeth — played in dinner jackets to jazzy accompaniment — has attracted some damning reviews and even some letters of complaint from the public. The letters have made Rylance realise that certain elements want “Tudor fundamentalists” at the Globe.

“We need to be more careful about letting them know what we are doing.” He says the Globe was always intended to be an experimental place; among this season’s developments is a week’s re-rehearsal period for each production, during which a director will spend an hour with each individual actor and take the work back to first principles.

A week after our conversation, Kerry Fox is quoted in another newspaper as being ambivalent about whether the sex in Intimacy was real. I catch up with Rylance on the phone: “I don’t know why she said that. Perhaps she was trying to preserve the magician’s trick of acting. Anyway, it is all an illusion. There is no real sex any more than when someone is killed in a movie they really die. It must appear that we’re both exhibitionists, but we were cast for our acting. The whole thing is a story. Pretend.”

Jealousy is a theme in both Cymbeline and Intimacy. Rylance sees his work as all-of-a-piece. “It will be nice if people who like the film come and see the play as well.” As for watching Intimacy with an audience, he jokes that it would be easy to escape recognition: “I could just wear clothes.”