The recent referendum on Britain's membership of the European Union sadly generated more heat than light: both the campaigns before and the aftermath of the vote for Brexit were marked by inane rhetoric and paranoid bombast. Fortunately, writer Afsaneh Gray remembers that theatre can be a valuable place for the exploration of complex ideas.

'I was inspired to write the play by a number of conversations I was having with friends of mine with mixed-race backgrounds,' she says 'We were getting fed up of being asked to represent our communities.' Being half Jewish and half Iranian, Gray recognised that conversations about racial identity ignore a salient fact. 'Mixed race is the fastest growing ethnicity in the UK. It struck me that our experience – the experience of those who don't quite fit into a white Britishness but also don't quite fit into anything else – is rarely seen or explored.'

Octopus evolved from these observations. A story of young women who fail to fit lazy categorisations, it follows the decision of Sarah, Sara and Scheherazade to refuse the question of Britishness and form a punk band. And in their self-definition as 'octopi' – not only mixed race but mixed up about their heritage – a new, emergent British identity is suggested.

Gray's attitude towards notions of Britishness and otherness is not merely a simple rejection of racist complaints about immigration. She has observed that the anxiety to include excluded voices has led to absurd situations. 'One friend, for example, had been invited into the rehearsal room of a play set in a country that her parents were from, but felt profoundly uncomfortable in the role of the authentic voice,' she reflects. 'First of all, her religion was different from that of the characters in the play. Second of all, she had never lived in the country of her parents' origin.'

With music by harpist Serafina Steer, Octopus is far from a dour polemic on the wrongs and rights of racial identity. Like its heroines, it rejoices in an anarchic wit that challenges ill-informed notions. 'As rhetoric around immigration and terrorism is becoming increasingly rabid, a new kind of anxiety over Britishness and national identity has crept in that makes our position feel less secure than it did when I was growing up,' Gray continues. It's in this mixture of passion and comedy that Octopus aims to hit home.

'I hope the audience will laugh, see themselves in the play. I think that's where a lot of laughter comes from,' she continues. 'They're very welcome to cry – somebody did at the Greenwich Theatre reading – but most of all I hope they leave with a sense of anger and purpose.'

Although political theatre is always common at the Edinburgh Fringe, Octopus is remarkable for taking on a topic that is controversial, and not easily resolved. The characters find a solution to the state's attempt to pin them down to a particular community – effectively making their own – but, as the referendum exposed, the current political climate is encouraging all manner of retrogressive attitudes. Gray's sense of injustice, and sense of humour, challenges these ideas, lending Octopus the same fierce freedom that its heroines demand through the power of punk.

Assembly George Square Theatre, until 28 Aug (not 15), £10–£11 (£9–£10).