They arrive on stage wreathed in dry ice, cast in shadows and dressed in impeccable black suits. Indeed, it’s hard to imagine the New York group any other way; they must walk to the bus stop shrouded in fog, and go to sleep in freshly pressed trousers and jacket. But the stern purposefulness of their entrance and the sharpness of their full-sized Ant Hill Mob get-up signal a maturity and determination to tap into the dark, turbulent but cathartic mood that’s served them well for more than 15 years – and proves to still have plenty of life in it tonight.

It’s a sound they coined on 2002 debut Turn on the Bright Lights, a windswept, ruefully melodic post-punk album that was pared back but possessed a widescreen ambition. These are serious men, you sense, and this is big music, tonight filling the cavernous Royal Albert Hall with a rolling thunder of artful angst.

Dapper guitarist Daniel Kessler seems to have a magical self-replenishing sack full of the oscillating, haunted and spare riffs that power their songs, like The Edge in his prime. Each one seems to spark off sentimental journeys among the faithful, Not Even Jail prompting strangers to bump fists and hug in the stalls, its every subtle tempo change feeling seismic.

Dapper ... Daniel Kessler. Photograph: Burak Çıngı/Redferns

But Interpol aren’t coasting on the nostalgia these songs prompt, but rather the very specific emotions they tap into, their seductive fictions evoking a certain generational dislocation and building it into something darkly glamorous: a romantic, understated, but very real sonic theatre. The tender, raw-edged burr of frontman Paul Banks’ doomy Gene Pitney vocals is ageing wonderfully; on All the Rage Back Home, he’s a cigarette and whisky away from being a wracked Sinatra, singing torch songs for shabby clubs that actually scrub up well in these more salubrious environs.

There’s alienation to this music, but the chord it touches makes it anthemic, and unifying. The regular comparisons they’ve enjoyed/suffered to Joy Division have their place tonight, not least in the purposeful, tension-ratcheting rhythms, tighter than Kessler’s drainpipes (though on Take You on a Cruise they tap into a razor-edged groove that sounds like a goth Chic). But Interpol don’t have Joy Division’s unearthly coldness, and Banks doesn’t channel Ian Curtis’s utter desolation. His voice frail and interesting, breaking up like a disintegrating phone line, he might sing of loneliness and disaffection on NYC, its imagery of pavements as grubby as pornography now perhaps a relic of a bygone, pre-gentrification Big Apple. But the effect is one of connection, of something universal. Interpol’s songs have edge, but they’re comforting; conjuring a darkness but saying: “You’re not alone in here.”

Their landmark, cornerstone tunes remain impactful, but some of tonight’s finest music comes from this year’s Marauder. Their sixth album marks a redoubling of intent, but also leavens the gloom a little with the perspective of age, and a melodic touch grown surer with experience. The Glitter Band stomp of NYSMAW quickens the blood, while the brash new wave chords and snake-hipped rhythms of Complications make for indie-rock with eerily keen aim, Banks’ greying croon now suggesting the warmth of latter-day Michael Stipe rather than the alienated spook of yore. It’s evidence that Interpol’s spectral intensity and thrilling emotional shadowplay is weathering, but wearing well, and as they conquer landmark venues like this, they continue a path towards the pantheon that seems unstoppable.