In the first few pages of The Drowned Detective, we think we know exactly where we are: three former Special Services types, now working as private investigators in an unspecified eastern European city, have been hired by a government minister’s wife to find his adulterous lover (who turns out to be a rather plain woman who runs a tyre-repair shop). This opening is vibrant and wholly engaging, so we happily settle in for a fast-paced detective thriller, complete with car chases, government corruption and labyrinthine betrayals, all spiced with world-weary banter à la Dashiell Hammett, or Len Deighton. But no: a few pages later we become immersed in the twisted but beautifully observed fantasies of a jealous husband who cannot shake off the sense that he is as much sinner as he is sinned against. Then the territory shifts again, plunging us into the kind of ghost story that Kafka would have written, had he been inclined to that genre. All of this serves to demonstrate that you can never know where you are going with Neil Jordan. He is a past master at leading us up – or more likely down – the garden path.

In 1993, Jordan remarked that his films “start with realistic premises and lead to seemingly unrealistic conclusions. And I am interested in the way politics, racial issues and sexual images impinge on that journey.” This observation describes The Drowned Detective perfectly. The premises are, indeed, realistic, sometimes even banal, but what Jordan does with them is extraordinary. Not much can be said about the plot here, partly because to do so would undermine its suspense, but also because it would risk spoiling the reader’s enjoyment of its genre twists and turns. What can be said is that the narrator, Jonathan, lives with his wife, Sarah, and his daughter, Jenny, in a city that is crumbling around them, its streets a seemingly never-ending battleground between the police (in black balaclavas) and Pussy Riot-style protestors (in brightly coloured balaclavas). As the novel opens, Jonathan has just discovered that Sarah has had a one-night stand, at the very least, with one of his employees. Even before that, however, it is clear that the marriage was in trouble, though it is hard to know what the underlying problem might be, other than the possibility that this city, always hot and humid, constantly in a state of flux, has somehow infected the couple. Now, emotionally weakened by jealousy and self-doubt (he is referred to, on several occasions, as “unmanly”), Jonathan takes on a missing persons case that will lead him into a surreal and terrifying realm in which the city becomes one immense hallucination and everything that once seemed real and solid is cast into grave jeopardy.

Alongside this highly suspenseful narrative, The Drowned Detective offers a host of deft characterisations, such as Jonathan’s colleague, Istvan, a plump man “with an owlish face and kind demeanour. What he most enjoyed was never getting to the point.” Or Jenny, a bright musical child who “functions very well … among imaginary things” and has not one but a whole coterie of imaginary friends, who join the family each evening when Jonathan serves dinner:

I’ve laid out plates, I told her, for Melanie and Jessica but I can’t remember the third name. Rebecca, she said. Will they all take Parmesan cheese? I asked. No, she said. Melanie’s lactose intolerant, Jessica and Rebecca are on a diet.

Sarah worries about these imaginary friends, but Jonathan seems to enjoy his daughter’s vivid imagination – until Jenny acquires a new friend who is not quite so innocent as lactose-intolerant Melanie.

The most intriguing character of all, however, is Gertrude, a psychic and self-confessed charlatan, whom Jonathan consults, not only for his missing persons case, but also in his efforts to find out what is happening to his marriage. Gertrude is a wonderful creation. A latter-day Marlene Dietrich with her Pomeranian lapdog, her cigarettes and her cocktail glass (which contains a mix of crème de menthe and wheatgrass), she becomes Jonathan’s oracle, uttering cryptic predictions and observations that, like the oracle at Delphi, mainly serve to bring her client’s own most secret desires and fears into the light. The Drowned Detective is a powerful study of the psychology of jealousy and a man’s fears of being judged and found wanting; it is also a book about fate and chance, and how, within the seeming apparatus of destiny, a form of grace is revealed by the suprarational powers of the imagination. That Gertrude is the instrument of that grace seems entirely fitting. Somehow, in a city that is visibly decaying, she rises above the chaos and, in so doing, offers her “rationista” friend a way to overcome his greatest weakness, “that English thing you call logic”.

• John Burnside’s Black Cat Bone is published by Jonathan Cape. To order The Drowned Detective for £12.99 go to bookshop.theguardian.com or call 0330 333 6846. Free UK p&p over £10, online orders only. Phone orders min p&p of £1.99.