In mid-twentieth century Japan, a family from Tokyo arrives in the countryside; a father and his two young daughters, 10-year-old Satsuke and four-year-old Mei. They're excited by their beautiful surroundings, but this isn't a holiday. They've come to be near the girls' mother, who's convalescing in the hospital. Little Mei is playing outside one day when she encounters two peculiar creatures, like fluffy little bears, or perhaps more like rabbits. She chases them into the deep, dark woods, where she encounters an extraordinary giant. All of this is Mei's vivid imagination, of course. Except that Satsuki sees the creatures too...

Well, they did it.

The Royal Shakespeare Company's new stage version of My Neighbor Totoro is running at London's Barbican Centre and it's… let's put it like this. There's a grumpy anime director called Hayao Miyazaki, and he's famed for monstering people who give presentations that offend him. And by "monstering," I don't mean a furry friendly monster who loves raindrops, acorns, and spinning tops. To raise the stakes even further, Totoro is Miyazaki's most personal film bar none, the one where he reclaimed his troubled, frightened childhood and made it into magic for the world. So it's no small thing to attempt to bring Totoro to stage.

Directed by Phelim McDermott and written by Tom Morton-Smith, this Totoro has many tricks, but the main one is how it pulls off two very different things at once. It feels extremely faithful to the classic 1988 film, and it feels joyfully, playfully creative, as if it's finding out what it can do even as you watch. It's funny. It's sometimes plain daft; it has moments that could be from a really good pantomime. But then the play can flip tones to something that's awesome, and scary, and even anguished. Rather than a panto, this Totoro feels more like a dance, a dance that sweeps up the two capering little girls, Mei and Satsuki, black soot creatures, timid rabbit-size critters, and even corn-filled hedgerows (which are whirled around by the play's chorus of puppeteers) and bits of houses that reconfigure on a turntable like giant-sized puzzle pieces.

For anyone who's seen the film, the Totoros are what you sit and wait for. How is the play going to do that scene, and that one, and that? But the play rightly foregrounds the girls, as it must do; all the wonder of the Totoros is channeled through them. And the two actors – Mei Mac as Mei and Ami Okumura Jones as Satsuki – are superb. You know they're adults playing, or more accurately performing, characters who are a fraction of their age, but they throw themselves into their parts with the same sense of discovery that the play has. They're funny, endearing, vulnerable, brave, bratty, and noisy. (A typical Mei line: "We saw a COW!") You don't picture them as the girls from the anime; you accept these are Mei and Satsuki on stage.

As in the film, Mei gets the biggest moments, not just when she's crawling over something's giant furry tummy, but later on too, when the magic is nowhere in sight and she's more hurt and scared than any child should have to be. Beside her, Satsuki can only try to lead her as a good big sister – the play amplifies a moment when Satsuki berates herself for letting Mei down, and it feels like a climactic moment from Mamoru Hosoda's Mirai, another story of young siblings and their overwhelming emotions. But even as Satsuki strives to lead Mei, the irony is that Satsuki's happiest moments are when she's following Mei instead, or when the girls are stepping out spontaneously together.

Among the supporting cast, there's the country boy Kanta (Nino Furuhata) – he's the one who stares at Satsuki from afar and acts like a preteen tsundere. He's mined for comedy, turned into the kind of lad who's terrified to say one syllable to a girl. There's a gem of a new scene, where Kanta is feeding his household's chickens, and somehow the chickens end up ganging up on him (it's a psycho-chicken scene!), and it's a riot. Kanta's granny (Jacqueline Tate) is rounded out, too. She's still lovely, but with a bit more temper and worry than the film gave her, and the added detail that she had a sister too, once. The girls' father Tatsuo (Dai Tabuchi) is close to the film, but with more humorous stress on how he depends on his responsible daughters to get him out to work in the morning and keep things running at home. Again, it feels like Hosoda's Mirai, and how the harassed at-home dad was shown there.

The humans, then, feel real. The Totoros feel as real as they need to be. I won't spoil the specifics of how the Totoro are done, but this never feels like a high-tech play. There isn't the ostentatious stage machinery or video screen effects that you get in Back to the Future: The Musical, to take another recent adaptation from film to play. But you do get huge creatures standing on stage, and they're quite something. Many of the play's effects show off exactly how they're being done as you watch them. But there's one sequence, which runs for what feels like a long time – certainly longer than the famous scene it adapts – that will have you thinking, how the heck are they doing that on stage?, and it's a piece of theatre you'll carry in your head long after.

The creatures often look goofy, but good goofy. The little Totoros pop up all over the stage like they're in an old Scooby-Doo cartoon. The giant creatures sometimes feel like they've emerged from a child's drawing (presumably Mei's), but that feels right, too. As for the expressions of the giant King Totoro, I was reminded of the first King Kong in 1933, which had moments when the filmmakers used “real” life-sized props to punctuate the stop-motion. Only now you're seeing giants on stage, right in front of you.

But the play evokes animation too, many times over. While I described the play's named characters, there's another set of performers. These are the black-clad, usually black-veiled puppeteers-cum-stagehands who are constantly on view, manipulating scenery, wielding the smaller critters or coming together in groups to move or lift the big ones. It's like a stop-motion behind-the-scenes film, when you see the animators tenderly moving the models they're acting through. On stage, sometimes the puppeteers are pushing and lifting the human characters, too.

They also double up as extra humans when needed, for instance, to play the kids in Satsuko's classroom. When they move scenery, sometimes the girls themselves see the scenery move, as in an arresting sequence where Mei is lost in hedgerows that spin around her, and even “attack” her. If this was animation, it'd be a more psychedelic cartoon than anything Miyazaki would do.

The play also goes beyond Miyazaki with prods at the fourth wall. A few effects go slightly “awry” for deliberate gags. A puppeteer has to be prodded by his teammate, because he's forgotten to switch into an in-world character to move the current scene along. If these jokes were animated, they'd be more Aardman Animations than Ghibli. The play has a couple of lovely moments of what are low-tech “animation,” pretty much, and they'll make older viewers remember the cut-out TV shows by Britain's Smallfilms studio, the makers of Noggin the Nog and Ivor the Engine.

Granted, the play makes choices that won't please everyone. There's one key bit of the film where you definitely wonder "How will they do it?" as the play approaches the moment. What the play actually does with this challenge is hilariously low-tech and very charming, but it left me disappointed, like a kid opening a present and finding socks. Or to make another animation comparison, I felt like I was watching an anime build towards a huge battle, and then the battle itself is just still images. This happened right before the interval, and left me feeling slightly flat... and then the second half started with a different low-tech effect, and this one delighted me so much that I forgave the play instantly.

Unsurprisingly, the play lengthens conversations that were in the film, and adds new ones. Some Japanese expressions are thrown cheerfully into the English dialogue, like "itadakimasu!" and yatta!. More seriously, Mei and Satsuki are shown as being very aware of tragic truths. It's Mei, I think, who first brings up the subject of death in the play, with disarming, unsweetened directness. Satsuki, meanwhile, knows all too well that adults lie about bad things, and she won't be lied to any more.

There's fear in the play, but it's mixed with awe and delight, especially when the huge furry figures loom from the dark background of the stage. The scene where Mei enters the deep, dark forest feels even more Alice in Wonderland than it did in the film, like she's entered an old woodcut illustration of a fairy tale. The departure of the soot-sprites from the country house turns into a quiet, measured dance of hairballs in the air. It's the play's longform equivalent of all those pillow shots in the film of snails on grass blades and leaves on streams.

Sounds of thunder, wind and torrential rain augment the experience. So does the orchestra, which is visible at the back of the stage, playing what are often variants of Joe Hisaishi's film score. It's not afraid to go louder than the film at times to add crescendos at moments of crisis. Sometimes a soloist (Ai Ninomiya) sings in both English and Japanese, most urgently when a child is missing and the play's darkest dance is being performed against the sound of deep, deep water.

I heard sniffles behind me during the play's later scenes, when the characters are at their most distressed, and that's how it should be. But at the end, what looked like the whole audience rose in a cheering standing ovation, with tricks and jokes all the way to the final curtain.

I saw one of the play's preview performances – press night is on October 18. I paid for my ticket, which wasn't cheap (£85), though I had the luck to snag an ideal stalls seat near the action. Don't go into this Totoro expecting a digital-era miracle like a Hatsune Miku concert. Do go for terrific acting and stagecraft and shared make-believe and copious amounts of stage fur. The biggest compliment you can give the play is that you can imagine it's the stage Totoro that Miyazaki would make himself.