My father loved Shakespeare, but thought he was infinitely better on the page than the stage. He disliked directorial interventions and feared the poetry would be lost by someone's smart ideas. The first Shakespeare play he ever took me to see was Cymbeline, possibly with the dastardly intention of proving that some of the most beautiful lines ever written, the song "Fear no more the heat o' the sun" would be wrecked by the actor saying them.

I can imagine, then, how he would feel about Matthew Dunster's radical production of the play, which not only renames it Imogen after the character with the most lines and the strongest moral gravitas, but also gives us only the first two lines of that famous lament, in grumpy tones, before sweeping on with the action.

There's no poetry in this production, radically reinvented, drastically cut. There are no gods or heavenly interventions either. Just a setting by Jon Bausor that swathes the Globe's stage in heavy plastic sheeting, puts the cast in tracksuits, and reimagines Cymbeline King of Britain as an urban gang leader, with a tattoo, fighting off the warring Romans, as brutal and sadistic as they come.

There is something to recommend this approach. The opening scene, with hooded gangstas dividing up the cocaine to the pulsing beats of grime artist Skepta is thrilling, and Christopher Akrill's street dance choreography, all swagger and flash, works pretty well though I couldn't see the point of the aerial battles. There's an energy here which revitalises a truly difficult play bringing to a conclusion Emma Rice's first season as Globe artistic director that has been characterised by taking some of Shakespeare's least performed plays and rethinking them bravely.

The greatest bonus is the way it liberates Imogen, the king's daughter who loves the – if we're honest – pretty wet blanket Posthumus with all her heart, to be a bold heroine, active rather than purely reactive. Former EastEnder Maddy Hill grabs the opportunity with both hands, stamping across the stage, shoulders hunched, full of purpose. She has terrific presence and – unlike some of the cast – the gift of making the language both modern and clear.

This terrific performance is matched by others. Matthew Needham's Giacomo, the schemer whose lies about Imogen's virtue set events in train, is a force to be reckoned with, ashamed even while pursuing his plots, popping beer cans, exuding charm. Joshua Lacey turns the Queen's son Cloten, unworthy rival for Imogen's hand, into a swaggering blonde-haired bad boy, with a running joke when he demands his hangers-on follow him, and they don't.

Best of all there's real tenderness and wonder in the scenes where – in a plot too convoluted to explain - Imogen is unwittingly reunited with her lost brothers now living in the care of Martin Marquez's kindly Belarius. He and Scott Karim's Guiderius communicate with William Grint's Arviragus in sign language, using the actor's deafness to make visible the bonds that tie them.

There's such poetry in those scenes, it seems unbelievably perverse not to throw in a bit of Shakespeare's best verse as well. We get a quote from Daft Punk but it would have been nice to have the bard and the band side by side. Because if you dispense with all the poetry and oddity in Cymbeline, you're left with just the plot. Which is hardly its strongest suit, even if that does make me sound like my dad. Dunster has thrown the baby out with the bathwater and his production is the poorer for it.

Imogen runs at Shakespeare's Globe until 16 October.