Four hours prior to her deportation, a young woman lies on the floor at Yarl's Wood Immigration Removal Men, bleeding. We never see her; only the young security officer, half a broken mug in his hand, standing outside her door, shellshocked and panicking.

Removal Men watches the institutional fall-out as two security officers and their aloof superior attempt to handle the situation. The three of them must stick to procedure and cover up a crisis to get this woman on a plane and out of the country. That the 'resident' is pregnant only complicates matters.

It's dreamier than that, though – less linear, more expressionistic. MJ Harding's script is strange and unsteady: a quasi-musical in which soft songs swim out of banal back and forth as colleagues shoot the shit. Out of that, Harding draws a sharp critique of a system that dehumanises just about everyone involved – not just the immigrants in its care, essentially incarcerated, but also the employees in charge.

George and Mo are partners, but we first meet them in a sex club; Barnaby Power's paunchy George urging his junior into the fray. We're always aware of the men beneath the uniforms, not least when George arrives on shift with a butt plug still in situ. They're contract workers in clip-on ties. Power gives the older man a cheerful carelessness, while Mark Fields plays his doltish, maudlin junior. They're bossed around by Beatrice (Clare Perkins), a busybody in a trouser suit, just astringent enough to emasculate and infantilise them.

Removal Men isn't just sharp on the crisis of masculinity, but on the sheer stupidity of the institutional system. Harding tugs at the tension between hierarchy and empathy: one dependent on shared humanity; the other, on differentials and deference. George and Mo routinely trip into classic double act patter, a quintessentially British comedy hinged on hierarchies, but elsewhere Harding's surrealist satire is more scornful. He knocks "empathy workshops" and savages the superficial compassion that masks institutional violence.

The music is a masterstroke. Jonah Brody's songs sneak up on the script. Speeches seem to snag on melodies, and shouts fall into rhythm to become barked, breathless raps. Lyrics repeat like guilty thoughts, the sort you can't shake off, and Brody's music is woozy and delirious. In mixing untrained voices with an autotuned sound, the effect is both human and not. You hear larynxes catching and notes quivering, but the voice emerges almost electronically.

It's not an easy watch. Sluggish and low-key, Removal Men sometimes goes slack, but Jay Miller's production holds its nerve and sticks to its guns. Its best moments are brilliant. Joshua Pharo's lighting flushes it with colour, then plunges back into institutional glare, while Bethany Wells' minimalist design slowly reveals its institutional colour scheme – a lick of yellow that suggests something hazardous. Harding proves as much: his institution would dehumanise us all.

Removal Men runs at the Yard Theatre until 11 December