But if you meet her, what you don't get is po-faced sobriety, even if her high spirits can't always spill out on to the stage. "You can have a good time with Masha in Three Sisters and we used to get a few laughs out of Nina in The Seagull but I cannot drag a laugh out of Anna Petrovna," at which point she dissolves into giggles.
Throughout our conversation, the cool landscape of her long pale face keeps shifting as different emotions pass across it. Small wonder that she has done such startling work on TV. Walter's break came as Cathy in Richard Eyre's TV film of Ian McEwan's The Imitation Game (with Bill Paterson). Eyre built much of the film on expressive reaction shots of this intelligent, innocent, well-bred young woman. It was a remarkably brave, moving performance, yet it didn't catapult her to big-screen success. Luckily for theatregoers, who are just about to enjoy the next instalment of her glittering stage career, as she and Paterson are reunited playing Chekhov at London's Almeida Theatre.
This time it's the early, rarely performed Ivanov, in which she plays Anna Petrovna, wife to Ralph Fiennes's Ivanov. Her character is pivotal but absent for much of the play. "It's frustrating already," she muses. "You have to build up in the wings for an hour, then get on their side very quickly and then disappear. If you keep going all evening you don't have time to question what you're doing. It's always harder to do these parts than those that go on all the time. Well, not harder... but if you've got a very large self-critical worm in your brain like I have, then you've got hours to either rue what you've just done or to plan something which then doesn't come off at all like you've planned it."
She's full of praise for David Hare's new translation. "I'm always nervous talking before a show has opened because everyone will say she was completely wrong, but I've seen the play a couple of times and never before felt so compassionate about Ivanov." She believes Hare and director Jonathan Kent have faced the pitfalls in this youthful piece and made it more accessible. "He's a man who's so unhappy - he says far worse things to himself than anyone could say to him. The audience cannot despise him more than he does himself. Everybody who tries to criticise him is punching thin air because, after a stream of abuse, he says: `Yes, you're probably right'. What's interesting is that he has the intelligence and self-awareness of a Hamlet but, within the play, repudiates the whole notion of Hamlet being romantic. There's nothing romantic or wonderful about someone who is so melancholy. It's almost a clinical portrait of a manic depressive so you get beyond blaming him and can't help but feel sorry for him. We all know you can't help it if you fall out of love."
This tragic element appeals to her but so does the vigour and youth of the writing. "It has all this anger in it and all this uncompromising head-on view of things. There isn't that web, that intricate veil that covers The Cherry Orchard or the others that are maybe more seamless, but I love that this has such highs and lows."
It's impossible to imagine your typical GP knocking out a steam of dramatic classics between house calls, but Walter sees the doctor in Chekhov coming through in the writing. "He's done such a good portrait of a man who was 10 years older than he was when he wrote it. To have that level of understanding might be the doctor in him. He made great clinical studies of people and was a great watcher of what was going on but I still find it extraordinarily understanding of a position which couldn't yet have been his. Although of course I was frightfully melancholic at 19."
Reading some of the less enlightened interviews in the cuttings file, you might imagine that she still is. "A single, 40-something workaholic," they tut, knowingly, unable to gossip about failed marriages or a torrid home life. "My private life is perfectly nice, thank you, but it's not that fascinating. I've always maintained you'd learn more about me if you talked about my work because that's where my heart beats." I remind her of a Daily Mail feature that sank to the level of asking her the contents of her handbag. "I decided the best thing to do was to make things up," she laughs. "I lied from beginning to end! There's a misconception that work is sad, narrow and boring. If you're involved in the theatre it embraces all sorts of huge things. That's very hard to communicate. I completely shrivel up and then I sound boring and pious."
All of which makes her a self-confessed workaholic? "I suppose I am, yes. Tedious, isn't it? Get a life, Harriet." Joking aside, she's expansive and bracingly intelligent about her work. There's a natural reticence about her - her graceful, slender hands are as self-protective as they are descriptive - but she understands her talent. She's not arrogant - she knows there's a huge amount of luck involved - but she's refreshingly free of bogus "who, me?" affectations. She turned down Hedda a couple of times because she wasn't convinced by the directors involved. When she finally agreed to play her, it was for Lindy Davies for whom she was a luminous Anna in the mesmerising production of Pinter's Old Times, a role she took on with only 10 days' rehearsals.
"I was very sceptical but there was something about what Lindy said in the interview. `We've all seen good acting. We're all expert at pretending. I'm interested in something else happening,' and I thought, that's it. I'll try that. If you've been at it quite a while, you can pull a few tricks out of the bag; you can impress people."
Davies, a former actress who suddenly succumbed to stage fright, has developed a method of working that recognises the process of acting. "Far too many people think we just get up there and do it because we're `instinctive creatures'," says Walter. "So many people in professional positions don't recognise that there is a process you can lock on to. There are directors who compensate for the lack of that by having brilliant taste or very good judgement and all sorts of other things which make you want to work with them, but..."
Walter, of course, is far too shrewd to name names but on wider issues she has always been vocal. Something she learnt when working with political companies of the calibre of 7:84 and Joint Stock at the beginning of her career. Much of that work harnessed not only her intelligence but also her comic skills. She brought the house down at the recent splashy Royal Court fundraising bash, reprising her beautifully timed Biddy from Three Birds Alighting On A Field which would have come as no surprise to anyone who saw her formidably funny Lady Croom in Tom Stoppard's Arcadia, a role she was terrified by. "I just read it and thought: Maggie Smith, Maggie Smith, Maggie Smith. She should be doing this." In 1980 she was in Joint Stock's revival of Caryl Churchill's Cloud Nine, a deeply touching comedy of sexual politics that she remembers as a tremendously witty play. "I love wit. It's such a weapon, it's such a conspiratorial instrument... when you can get the audience and then you think, `Ah, you've got it, great, we're together'." The words ring with passion. Her eyes are wide with relish. "I love it"
`Ivanov' is at the Almeida Theatre to 5 April (0171-359 4404)