Earlier this year, around the election, there was a crowd-funded poster campaign. Professionals of all sorts - doctors, teachers, comedians - stood in their work clothes under a slogan that read, "I am an immigrant." Admirable though it was, the campaign was less about people than their value. Mohammed Taj, a bus driver born in Kashmir, was little more than a smile and a well-tied tie. Lukas Belina, a Polish fire fighter, was a nice face and a fireproof suit.

Inua Ellams goes further in this improvised autobiography - a one-off that may yet become a work-in-progress. He wears the label 'immigrant,' but he also resists it. I'm so much more, he seems to say: not just a poet, not just Nigerian, but an entire life's worth of people; a kid who got bullied, a teenager using basketball to boost his sex appeal, a young man who found poetry through soft-porn on Channel 5. Ellams intersperses his story with his poetry: a retrospective re-arranged to shed light on a life.

It starts in Nigeria, with a baby boy born unexpectedly (he hid behind his twin sister in utero), before leaving for London, then, briefly, Dublin. All the way through, Ellams threads in images that shed light on immigration, painting a picture of a patchwork identity: born between star signs; Muslim father, Christian mother; a boy that played with Barbies and ate Nigerian food with chopsticks. People and places infuse into and inform his identity - as they do all of us.

Ellams is not a trickshot poet and not, like Kate Tempest, one to go into a trance. Instead, his writing is aromatic. He can capture the smell of a place, the sense of a moment, as well as anyone - be it the parched backyard of his childhood, where he killed lizards for sport, or the drab Dublin school that came later. And yet his narratives are so easy to follow, they might as well walk through the streets holding an umbrella in the air.

'Even when onstage at the National Theatre, Ellams was technically an illegal immigrant'

His account of emigration - because, from Ellams' perspective, it was as much about leaving as arriving - is vital. After his father converted to Christianity, the family home was burned down by Islamists and, too proud to seek asylum, his father found a job in London. "No one leaves home unless home is a shark's mouth," he says.

The family followed - only for the Royal Mail to lose their passports and leave them in limbo. You realise the vulnerability in bureaucratic systems, where people become case files and forms, and the scale of sacrifice involved. This family that could send their son to boarding school in Nigeria are suddenly poor and outsiders. Racism is a new experience for them.

The night starts to soar as Ellams hones in on his point: that poetry was his lifeline. Trite, until you realise that, even when onstage at the National Theatre, Ellams was technically an illegal immigrant, battling through the legal system by day for the right to stay. He was an illegal immigrant even when he got an invite to Buckingham Palace for a drinks reception. Art - and its community - provided a home.

Part of the pleasure, tonight, is the gig's off-the-cuff manner. Ellams freestyles and overruns in a room filled with his friends, many of whom crop up in his stories. Should it solidify into a full, fixed piece, though, An Evening With an Immigrant will have real currency. With immigration such a political battleground, this is a necessary reminder - gentle, but firm - of what it really entails, and Ellams' account disrupts the prevailing insiders' perspective to provide the immigrant's perspective that often goes unheard.

An Evening With an Immigrant played at the Soho Theatre on 20 July