Such plays constitute a mini-genre that is quite distinct from biographical plays like Pam Gems's Stanley, or historical dramas like Richard Nelson's The General From America. The playwrights drawn to it are too diverse stylistically to be bracketed together as a movement; but each uses the same simple question - "What if X had met Y at a certain place and time?" - as the starting whistle for a game with history.
"The possibilities are endless with imagined meetings because you can play with the whole of history," says David Pownall, whose 1983 play, Master Class (not to be confused with the forthcoming bio-play about Maria Callas), has Stalin and his cultural supremo, Marshall Zhdanov, joining Prokofiev and Shostakovich in a room at the Kremlin one evening in 1948. Stalin tries and fails to coerce the composers into providing the uplifting music that he insists the Russian people need.
"Those four were all in Moscow at that time," Pownall explains. "I put them together to create a debate that would show, through Stalin's failure, how even the most autocratic power cannot defeat artistic individuality."
Master Class was revived in Romania in 1996 and Pownall returned to its format earlier this year with four 20-minute comedies for Radio 3, which featured imagined but, again, historically feasible, meetings between George III and George Washington, and Winston Churchill and Salvador Dali.
"I love using historical characters because in my imagination they start creating arguments that I'd not thought of when I first put them together," he says. "Shakespeare created countless imagined meetings to serve his dramatic aims in the history plays. I think that is still very much the business of dramatists: to treat history as a malleable form of time."
Stephen Churchett, the actor known to EastEnders fans as Phil and Grant's lawyer, echoes Pownall when he suggests that Tom & Clem, his first full- length play, "re-invents history to make it more dramatically accessible". Tom Driberg, the flamboyantly gay socialist who was the Daily Express's original William Hickey columnist, did report on the early sessions at Potsdam, just before his election as MP for Malden, but was back in England by the time Clement Attlee reached the conference on 28 July, 1945; Tom & Clem has them entering the press room on the same day.
"I'll probably get letters from irritated members of the audience," Churchett predicts. "But this imagined meeting has enabled me to bring together two real archetypes: the Cavalier, romantic Driberg, and the Roundhead, pragmatic Attlee. "
Fine performances from Gambon and McCowen in the play's try-out run in Guildford impressively fulfilled Churchett's hope that Tom and Clem's differences would "make for good drama". Against a sub-plot involving a Russian military attache, the politicians' argument about socialism contains deliberate echoes of the gulf between old and new Labour which give this customised view of history a powerful present-day frisson; though when Churchett completed the play in 1995, he could not have imagined that a week before its West End premiere opinion polls would be forecasting a repeat of Attlee's 1945 Labour landslide.
Where Churchett took his leading characters from 20th-century politics, Michael Meyer looked back to 19th-century theatre. The leading biographer and translator of Ibsen and Strindberg had "always been nagged" by the fact that these two theatrical giants had never met in real life.
"In 1885, bad weather forced Strindberg to return home while he was on his way to visit Ibsen in Rome," Meyer explains. "When his marriage broke up later that year, he became a misogynist and Ibsen was suddenly the great enemy for having encouraged women to leave their husbands through A Doll's House. Strindberg no longer wanted anything to do with him." Meyer plugged this historical gap with Meeting in Rome (broadcast on Radio 3 in 1991 and later staged in London), in which the dramatists discussed women, plays and politics, "the topics I reckoned they would have talked about".
Anyone who saw the recent Donmar Warehouse revival of Terry Johnson's 1983 play, Insignificance, or the 1985 film version directed by Nicolas Roeg, will recall the glorious initial shock when Marilyn Monroe walks into Albert Einstein's New York hotel room; yet their meeting is not all that improbable.
"Einstein was top of the 'Men I would most like to sleep with' list which Marilyn once made, and he was in New York in 1953 when she was filming Seven Year Itch," explains Johnson. "It therefore seemed reasonable for me to say 'She'll meet him and we'll see what happens'."
The dialogue which develops around Monroe's childlessness and Einstein's guilt over his contribution to the invention of the atomic bomb does not seem in the least contrived. With Hysteria, premiered at the Royal Court in 1993, Johnson built an equally convincing, if more fantastical, scenario out of the afternoon in 1938 when Salvador Dali took tea with Sigmund Freud in Hampstead. "With both plays," he says, "I was trying to reflect issues that touch us all by using the icons that we all know."
Unprompted, he adds that depicting legendary figures was less demanding than writing plays, such as Dead Funny, in which all his characters are fictional. "With Monroe or Freud, you start with an image and a set of presumptions that the audience shares. You've virtually eradicated the need for a first act."
Peter Whelan, who imagined a conversation between Shakespeare and Christopher Marlowe in School of Night at the RSC four years ago, agrees: "If you can plunge straight into a well-known image you are halfway to reaching your audience."
That's only half the story, counters Pownall. "I think these plays are harder to write, not just in terms of the research they require, but because you have somehow to embody in a short space of time the potency that figures such as Stalin have within history." Only if a writer portrays Stalin as a caricature, Pownall argues, can he enjoy the same degree of freedom as he would when creating a more ordinary Joe.
With so much talk of playing with history, it seemed reasonable to invite entries for a Fantasy Theatre League, in which punters choose the two figures they would most like to see go head-to-head on stage. Whelan suggested Churchill and Nye Bevan, "sworn enemies and the most powerful debaters parliament has seen"; Meyer opted for Jesus and Judas. Or how about Orson Welles and William Randolph Hearst enjoying a chat after a private screening of Citizen Kane? The possibilities, as Pownall suggests, are endless.
! 'Tom & Clem' opens tomorrow at the Aldwych, WC2 (0171 416 6007). 'Insignificance' will be broadcast on Radio 3 on 15 June.