Welcome to Waitsville. A place where bad jokes are good, Vaudeville never died, and the talk is of smoking monkeys, weasels and the mating habits of the preying mantis.

Few musicians have built their own world as resolutely and completely as Tom Waits– that twilight carny town, pitched on a scrapheap in some flickering no-man’s land between Beat generation Manhattan and the Weimar Republic. Entering the theatre in a sweltering Edinburgh tonight, though, the sense of crossing some kind of border into another place, with its own laws, is more intense than ever.

In the same way that “free earplugs” became one of the most common talking points surrounding My Bloody Valentine’s recent tour, the ultra-stringent anti-touting measures in place for Waits’s two nights in Edinburgh, his only UK shows, have already provoked endless pre-show chatter. Outside the Playhouse, we line up like patient refugees while a man with a bullhorn orders us to have not merely our tickets, but also our photographic ID and credit cards ready for inspection. The fact that a great many are actually producing their passports at the door heightens the feeling of shuffling through some weird Checkpoint Charlie.

Inside, the faded Victorian grandeur of the Playhouse somehow seems a little grander, a little more faded than usual. The waiting stage lies bathed in a mix of deep blue, hard gold and intense red light that lets you take in the impressive amount of percussion instruments at the ready, as well as the weird, mushroomy assemblages of 1940s-era loudspeakers. The PA plays a sublime mix of music that suggests old radio waves bouncing back off satellites – jump blues, country, R&B, early rock ‘n’ roll, garage rock, gut bucket gospel. Still, it doesn’t distract from the fact that heat is already tropical, and by the time Waits has kept us waiting 30 minutes past the strictly appointed 8pm start, the buzzing, sweating crowd is growing restless. Slow handclaps break out, minds perhaps beginning to turn to the average £100 ticket prices. (An unconfirmed, but plausible story: when, off the record, a representative of the agency behind the tour was asked why tickets were so expensive, the reply came: “That’s just Tom Waits having a laugh.”)

Any hint of ill will, though, is instantly dispelled when Waits finally appears and mounts his private little dais at the centre of the stage while his five-man band take positions around him. Before he’s even opened his mouth, he’s accorded an ovation, which he shamelessly milks and orchestrates, standing in the spotlight and raising his trembling arms like a sideshow conjurer, directing the applause. As he kicks into a ferocious bastard blend of “Lucinda” and “Ain’t Goin’ Down The Well,” the crowd is already fully hypnotised.

And “kicking-off” is right. From the first, the lead percussion instrument is Waits’s stomping motorcycle boot. As he pounds the beat out, you realise the podium he’s standing on is coated in a layer of white powder – flour, perhaps, or talcum powder, or maybe ground-up bones – so that a cloud of pale dust rises around him as he stamps. In his weather-beaten black suit and bowler hat, he’s part silent movie slapstick clown, part Samuel Beckett ghost, part New Orleans voodoo funeral director. He never breaks from his patented “Tom Waits” character, but his ragged stagecraft is flawless. Sometimes he jerks like a machine-man; sometimes he rattle about like a zombified bag full of bones; often, he stretches and moves inside his music with the languid grace of a ballet dancer, walking between raindrops.

The hammering industrial polka is maintained into “Raindogs,” before he shifts gears abruptly with an unspeakably gorgeous “Falling Down.” On songs like this – and, later, a similarly bruised and beautiful “Bottom Of The World” – Waits and his band suggest the kind of music The Rolling Stones might have made if they’d ever allowed themselves to grow up. But, as they remind you by lurching straight into a delirious “Cemetery Polka,” crashing from marching mutant oompah into lyrical, looping, drunken sweeps and back, the Stones never took anyone on a trip like this.

Waits’s band – Larry Taylor (bass), Patrick Warren (keyboards), Omar Torrez (guitar), Vincent Henry (sax, harmonica) and Casey Waits (Tom’s son, on percussion, and sometimes joined by his younger sibling, Sullivan) – define the term “sidemen.” They settle into the shadows, anonymous figures almost, but their instruments become lead characters, storytellers as spellbinding as Waits himself. Together, they lead you through the backstreets, bars, junk shops, jazz dives, civic halls, cafes and fleapit theatres of a mongrel city, a place where the German, Italian, Spanish, Jewish and Gypsy neighbourhoods all merge into one. A place where Mariachi bands invite klezmer clarinettists to sit in for wedding gigs in Harlem, where guitars can sound like zithers, and where Lester Young or Lightnin’ Hopkins might stop by any minute, looking for help on a spaghetti western soundtrack.

Meanwhile, in his dusty halo, their leader is singing, growling, roaring, hissing and whispering as if, after years of distressing it, he’s finally got his voice the way he always wanted it, sounding like Louis Armstrong and Captain Beefheart getting together with a failed opera singer to sing Billie Holiday in 1930s Berlin.

Picking up his guitar to thumb out the thick, itching groove of “Get Behind The Mule,” Waits takes us from cabaret to roadhouse, with Henry wailing out waves of distorted mouth harp that would have astonished Howlin’ Wolf. A grotesque, meandering tale of circus freaks resolves itself into “Table Top Joe.” For “Jesus Gonna Be Here,” we’re in a tent by the muddy river, listening to a rogue preacher testify his customised gospel.

Then, moving to the piano with only Taylor’s stand-up bass for support, the mood shifts again, the theatre transformed into a tiny, late-night barroom, as Waits begins to ramble and rasp out more and more of his trademark showbiz-barfly shaggy dog stories:

• “Y’know what annoys me? When people say, ‘My cell-phone is a camera, too.’ I mean – why can’t something just be what it is and be happy about it? Makes me want to say, ‘My sunglasses are also a tricycle.’ Yeah, but I never do.”

• “Y’know why shrimp never give money to charity?” (“Why?” comes the crowd’s foolish cry) “Yeah. Well, basically – they’re shellfish. (Pause.) My wife told me never to tell that joke again.”

• “I met someone who’d been there, and he told me what the moon smells like. Y’know what the moon smells like? Fireworks. Makes perfect sense. I mean, that’s where they all end up.”

And on and on he goes, dishing out patter like a washed-up Catskills entertainer. Of course, in between all the non sequiturs, he’s busy breaking hearts, breathing out exquisitely tender readings of some of his loneliest, most beautiful songs: “Picture In A Frame”; “Invitation To The Blues”; “House Where Nobody Lives”. During “Innocent When You Dream,” he leads the audience through a mass sing-along, the place sounding like a wartime music hall.

As he rises from the piano and makes his way back to his little stage, you can see that Wait’s thick black suit is drenched through in sweat. There’s just time to register the fact that, all around, people are sitting forward, literally on the edge of their seats, before he pushes the pace again, the group slamming into bone-shaking “Lie to Me,” and an immense “Hoist That Rag,” a long, fast, snaking, hard mamba.

“Hang Your Head” is burning blue soul. “Green Grass” manages to be sinister and gentle at once and finds room for a great whistling solo. On “Way Down In The Hole,” Waits mounts a successful, swampy campaign to reclaim his song from The Wire soundtrack, nailing down the beat by stamping on a fire bell and beatboxing like an asthmatic Rolf Harris. “Make It Rain,” follows, the band hammering out a slow, jagged pneumatic funk groove that builds mercilessly while Waits jerks and cries for his rain. It finally comes, when, in a flourish of pure, old school showbiz, a cascading shower of golden glitter suddenly pours down upon him from the rafters.

With the crowd on its feet again, baying again, Waits returns for a too-brief encore, laying out the sinister comic-book rumble of “Goin’ Out West,” and a weary, looping end-of-the-night waltz through “All The World Is Green”. Then, after two-and-and-half hours of booglarizing, beguiling and binding us in his unique spell, he’s gone, taking his world with him, and leaving us trying to find our way back to ours.

Astonishing, simply. And, in case you were wondering, worth every penny, and all the problems of passport control. Nights like this, performers like this, are as rare as rocking horse dung.

DAMIEN LOVE

SETLIST

Lucinda/ Ain’t Goin’ Down To The Well

Raindogs

Falling Down

On The Other Side Of The World

Shoot The Moon

Cemetery Polka

Get Behind The Mule

Cold Cold Ground

Table Top Joe

Jesus Gonna Be Here

Piano set:

Picture In A Frame

Invitation To The Blues

House Where Nobody Lives

Innocent When You Dream

Lie to Me

Make It Rain

Bottom Of The World

Hang Your Head

Green Grass

Way Down In The Hole

Dirt In The Ground

Make It Rain

ENCORE:

Goin’ Out West

All The World Is Green