"THIS isn't ballet ballet.

There's not a tutu in sight." In a couple of hours the lights will go down at the Kennedy Centre in Washington DC for the penultimate performance of Scottish Ballet's acclaimed adaptation of A Streetcar Named Desire. Dancer Arabella "Minty" Wraith, who plays Blanche Dubois, is reassuring me that it ain't Swan Lake.

"I've never seen anything like it. It's part play, part ballet, part physical theatre," she says. "The script is amazing, so to take the words away and still be able to tell that story so well, and so clearly, just through movement ..." She tails off, in awe. "Audiences just can't get enough of it. It's been standing ovations everywhere."

The idea for a dance version of Tennessee Williams' Pulitzer Prize-winning play came up in a conversation between theatrical director Nancy Meckler and choreographer Annabelle Lopez Ochoa. Scottish Ballet was already known for performing a diverse selection of works. Could it take on an American classic, revered for its characters and dialogue? Could the dancers kick pantomime habits ingrained since childhood and learn to act?

There had been Streetcar ballets before. A production created by Valerie Bettis for the Slavenska-Franklin company in 1952 is sufficiently obscure that nobody in Scottish Ballet had heard of it when I brought it up. John Neumeier staged a version in Frankfurt, in 1983, that is remembered for its challenging score. Nobody had yet managed to marry the drama to the dancing without compromising one or the other.

The first workshops were a struggle. Before Lopez Ochoa choreographed a step, Meckler worked intensively with the dancers. "They're often doing things like Alice in Wonderland and Sleeping Beauty, with very two-dimensional characters," she says. "We didn't want that kind of gestural pantomime, when people pretend they're talking to each other." One ballerina told her: "You have to understand, Nancy, we've been taught to gesture. It's much harder for us to work from inside."

The key to the adaptation was realising that the plot devised by Williams would have to be radically restructured. In the original, Blanche's back story is revealed gradually. Much of the action takes place off stage. Meckler and Lopez Ochoa decided to arrange the play's events chronologically, whether they are seen, referred to or implied in the text, and to show the audience everything.

That means starting at Belle Reve, the mansion that Blanche loses to creditors, causing her to move to New Orleans. The corp de ballet's first scene is Blanche's wedding to Allan Gray, a closeted homosexual who commits suicide after his infidelity with a male lover is discovered - events too shocking to even be hinted at in the screen version starring Marlon Brando, released in 1951.

"We thought, 'When she is haunted by these things, the audience will already have experienced it with her,'" says Meckler. "And when we put it together, we realised we were actually telling the ballet from Blanche's point of view, which doesn't always happen when you watch the play. Sometimes you feel like you're getting the story almost from Stanley's point of view."

This is partly Brando's fault. He only won the role of Stanley Kowalski in the original Broadway production when John Garfield made impossible demands to director Elia Kazan, but he immediately made it his own. In the script, Stanley is a bully who uses violence and threats to control his wife Stella, Blanche's sister, but the young Brando was so sexy, and so charming, that audiences often took his side, finding his put-downs of Blanche funny.

"When you read the play, you realise that Tennessee Williams wanted you to be horrified by his violence," says Meckler. "He rapes Blanche and he hits his wife, and all are done off stage so they don't always resonate with an audience. If you start off with Blanche's story you have much more sympathy for her by the time she arrives in New Orleans."

At Scottish Ballet, the roles of Blanche and Stanley were created by Eve Mutso and Erik Cavallari for a tour of Britain in 2012. The show came to the United States the following year, playing smaller theatres in smaller markets than this current trip, which has passed through Chicago, San Antonio, Houston, Charleston and Pittsburgh before reaching Washington DC.

In the performance that I watch, Stanley is played by Chris Harrison, a 32-year-old from Kippen, in Stirlingshire. He is currently the only Scottish dancer in the company. It's hard to reconcile the monster on stage with the softly-spoken dancer in the dressing room. "Obviously Stanley's a bit of a brute," he says. In his last scene, he rapes Blanche, sending her over the edge into madness. "Even though you can be happy about how it went, you're still not going to come off and say, 'That was awesome.'"

The other star of the show, although he'd bridle at the description, is Scottish Ballet's technical director George Thomson, an Aberdonian with four decades of experience in the theatre. From the start, he's had overall responsibility for the tour, from the logistics of making it run smoothly to the production design, the sound and the lighting.

Four years ago, when the show was first in development, he helped design the beer crates that make up the set - "there were all sorts of prototypes: to get the sound right, to get the weight right. They have to be robust" - and it's up to him to make sure they fall in exactly the same way each night when Belle Reve collapses. He has "11 really experienced, really good folk" on his team, and trusts them completely.

Although he's far from boastful, Thomson is evidently very proud of the show. For this run, he first came out to the States a year ago, to scope out venues, measure stages and work out how to transport everything around with a minimum of fuss. The whole production came across the Atlantic in a single shipping container and fits into one 55ft lorry trailer, but still manages to fill the Kennedy Centre stage.

An hour before curtain up, Thomson takes me on a tour of the theatre. In the basement, wardrobe mistress Mary Mullen is preparing the costumes, many of which have to be laundered between shows, and lamenting the fact it wasn't possible to bring the company's own twin tub washing machine and dryer on tour. In Charleston there was only one small washing machine at the theatre, forcing her to rely on the kindness of strangers.

"In Britain, absolutely everything comes with us," says Thomson. Because of the difference in mains voltage between Europe and the US, he's been obliged to work with the lighting equipment at each venue, but even that is agreed down to the last coloured filter. "You hear that ticking noise above you? That's some of the lights counting their sprockets," he tells me as we walk on stage, where set dressers are assembling the wall of crates, painted to look like a neo-classical Southern estate.

After briefly chatting with stage manager Zoe Hayward, who calls the cues each night, based on a musical score annotated in minute detail, we head up to the lighting and sound box. Lighting designer Matt Strachan shows me a few of the 200 lights, which each have 28 controllable attributes - "colour, position, shape, diffusion ..." - and the 28 old-fashioned light bulbs that fly in and out on a rigging system of his own devising.

Sound designer Gavin Jenkinson sets off camera flashes and gunshots for my amusement and talks me through some of the 100 manual cues and 80 sound mixes per show. Most of the performances on tour have been to tape, but the Kennedy Centre insisted on a live orchestra, adding an extra layer of complexity to an already fearsomely intricate production.

The sound and lighting mixes are pre-programmed - you would need to be superhuman to do them live. "It's not programmed so you press a button at the beginning and go and have a cup of tea. It's broken into bite-sized pieces in case something slows down or goes wrong," says Thomson. "It's a bit like flying a plane: it's sort of on auto-pilot for a lot of the time but you always need to step in and make adjustments."

I wonder aloud if there's a danger that I'll get distracted by the technical wizardry at the expense of the dancing. "I don't think you will," says Strachan. "I think the show will catch you."

He's right. The production is mesmerising from the start. Driven on by Peter Salem's score, which combines traditional dance forms with jazz and fragments of industrial electronica, the dancers convey the drama with breathtaking skill and grace. I'm no ballet expert but can't imagine connoisseurs being disappointed by the duet between Blanche and Allan that kicks things off, or the beautifully choreographed wedding dance, undercut with a nagging, discordant piano figure every time Allan glances at his soon-to-be lover.

Wraith is a compelling presence, fluttering like a moth beneath a bare lightbulb at the beginning and end of the performance, and alternately voracious and vulnerable in-between. The set pieces, at a bowling alley, a cinema and a nightclub, where Blanche drains one cocktail after another, are handled with aplomb by the ensemble. The one duff note is Stanley's cry of "Stella" after he's hit her, which elicits an unwelcome laugh from the audience, but it's hardly Harrison's fault - when The Simpsons have done a version of Streetcar, there's no easy way to play it straight.

The last three scenes, in particular, are hauntingly powerful. Blanche's retreat from the real world is symbolised by a vivid hallucination, in which the many men she has seduced and been seduced by push her around the stage in a bathtub, to the strains of a swinging version of Paper Moon. Then the stage clears, leaving her alone with Stanley. The stunned silence in the theatre as he rapes her is absolute.

For the finale, Meckler and Lopez Ochoa have transformed the blind Mexican flower seller - "flores para los muertos" - into a mocking cast of onlookers with carnations in their mouths. By now Blanche's fluttering resembles the delirium tremens of an alcoholic. As the curtain falls, the audience rises to its feet, offering heartfelt ovations to Bethany Kingsley-Garner, as Stella, and Remi Andreoni, as Mitch, in addition to Harrison and Wraith.

Immediately afterwards, I rush downstairs to meet Christopher Hampson, Scottish Ballet's creative director, responsible for commissioning new work and ensuring the standard of everything the company puts out. He's understandably beaming, with just one performance to go of a highly successful tour.

"It's huge being here," he says. "The Royal Ballet are here the week after us. Chicago and Washington and Houston are incredibly important cities in the arts. And the management here are absolutely thrilled with us."

Back in Scotland, the company will begin rehearsing Hampson's version of Cinderella. It still pays to perform the classics and Sergei Prokofiev's score is one of the most beloved in ballet. But as with everything Scottish Ballet does, it will have a modern twist.

Lopez Ochoa is choreographing Les Liaisons Dangereuses and Meckler is working on a dance version of The Seagull by Anton Chekhov. Matthew Bourne's adaptation of William Golding's Lord of the Flies has played to packed houses since its debut at Glasgow's Theatre Royal. Other choreographers have recently tackled Arthur Miller's The Crucible and The Handmaid's Tale by Margaret Attwood.

During the curtain call, when Wraith runs on to take a bow, a voice two rows behind me shouts: "Bravo, Minty." Her parents have flown over to see this, her first ever principal role, on one of America's most prestigious stages. She joined Scottish Ballet a year ago hoping that she might earn the chance to take dramatic risks there that she wouldn't at other companies.

"It is quite emotionally draining, especially the second half when Blanche descends into her mad world," she says. "If you really get into it, the rape scene can feel so real, but then that's what being an artist is."

The Sugar-Plum Fairy might not approve, but a generation of choreographers is taking note.

Tickets for Scottish Ballet's autumn season double bill of Elsa Canasta and Motion of Displacement are on sale now. Visit scottishballet.co.uk.