The words “back by popular demand” aren’t often applied to modern operas, except perhaps in ironic jest. Yet popular demand, as well as thumping critical approval, has brought Thomas Adãs’s magical musical take on Shakespeare’s magical musical isle back to Covent Garden just three years after the Royal Opera premiere.

In the interim The Tempest, written by the English composer when he was 32, has been acclaimed on the Continent and in the US. It’s a genuine hit, and you can’t say that about many recent British operas.

Monday’s superb revival — with most of the original cast back, sounding even more assured than after the scrambled preparations in 2004 — confirmed what the premiere had suggested: that this work is an ingenious melding of the familiar and the startlingly original. That applies not just to Adès’s music but also to Meredith Oakes’s terse, tangy, rhyming rewrite of Shakespeare. As in Purcell’s Fairy Queen and Bernstein’s West Side Story, the Bard’s words get short shrift. And the plot, too, is subtly tweaked so that the love of the youngsters, Miranda and Ferdinand, much more overtly confounds the powers of Simon Keenlyside’s immensely sung, tyrannical Prospero.

But it’s the eerie, airy music that ravishes the senses and drives the drama forward. Here too there are recognisable things: old-fashioned arias; cunningly weaved ensembles — none better than a sumptuous Purcellian passacaglia of reconciliation and renunciation near the end. But they exist in new sound-worlds that define characters and contexts with diamond clarity.

Best of all, the music seems to bloom in symbolic reflection of the redemptive love seeping through the play, at least as interpreted here. After the jagged shards of brass in the storm, the baleful orchestral grunts under Prospero’s vengeful lines, and the bizarre, stratospheric squeaking of Cyndia Sieden’s brilliantly demented stick-insect Ariel, Adès conjures a succession of meltingly lyrical set-pieces. Underpinned by ecstatic string trills there’s a gorgeous lovers’ duet, radiantly sung by Toby Spence and Kate Royal, making an ardent if somewhat consonant-free Royal Opera debut. Then comes a beguiling jingle of bells for the banquet and a shimmeringly nostalgic solo for Ian Bostridge’s rueful Caliban (a wonderful creation, like some fading heavy-metal headbanger turned scarecrow). Finally there’s a wistful fading lullaby for Caliban and Ariel that poignantly evokes the feeling of little lives being rounded with a sleep, even if the actual line is never sung.

Tom Cairns’s production, played amid the glowing cubes, flying gymnasts and basking alligators of Moritz Junge’s surreal set, still looks look like a bad night in an Ibiza disco. But with fine supporting performances from Philip Langridge’s anguished King of Naples and Donald Kaasch’s wonderfully malevolent Antonio, and with the composer himself conducting what really did sound like Shakespeare’s “thousand twangling instruments”, this is a show which gives one hope that an art form as old as The Tempest itself has been dazzlingly remade for the 21st century.

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