The Bourne Identity (2002) ended with our amnesiac hero, Jason Bourne (Matt Damon), discovering who he was and walking away from his old life as the CIA’s top assassin. Since then, Bourne has dropped out, pulled the plug on his past and gone off the grid. Or, at least, he’s tried to. In The Bourne Supremacy, Jason and his girlfriend Marie (Franka Potente) are living on a beach in Goa, India. It’s the setting for a perfect life, but Jason carries his murderous past with him like a piece of shrapnel lodged in his memory: come the night, it oozes the pus of bad dreams.

Meanwhile, over in Berlin, a clandestine CIA operation is in progress. Two CIA agents are trying to buy files that promise to reveal the name of the CIA mole responsible for stealing $20m of agency money. Pamela Landy (Joan Allen) and her team are watching the transaction on monitors from the safety of their office. Suddenly, they lose visual contact and hear shots; the two agents are dead. A fingerprint is found at the scene of the crime, belonging to Jason Bourne. After much debate, a CIA head comes to a decision concerning what to do: “We’re going to find that son of a bitch and take him down.” For the frame-up to work, Bourne has to be eliminated. So a Russian assassin is sent to Goa to do the job. He fails. Bourne assumes it’s the CIA up to their old tricks again. He doesn’t say it, but you can see what he’s thinking: I’m going to find those sons of bitches and take them down.

And so the stage is set for a spy vs spy drama — or maybe that should be spy vs spies. For the odds are heavily stacked against Jason: he’s outnumbered, outgunned, outgadgeted and outcomputered. From an anonymous building in Berlin, Landy and her team set out to track him down. Watching them at work, you get the impression that if Bourne were to break wind anywhere in the world, they could have a team of gun-toting agents there in five minutes to bust his ass.

The recent intelligence failings of the CIA concerning Saddam and WMD make the idea of the all-knowing and all-seeing secret service seem an implausible anachronism left over from the cold war paranoia in the novels of Bourne’s creator, Robert Ludlum. Likewise, the frame-up that brings Bourne back into action has no plausibility. Think about it. Would an agent as professional and thorough as Bourne leave one fingerprint at the scene of the crime? In the first film, Bourne was like a mad housewife, forever wiping everything down lest he was betrayed by a fingerprint or a pubic hair. What happened that night in Berlin — he forgot to wear gloves, or maybe one of the glove fingers went missing? Some of you might say this is the objection of an anal film critic who takes these things far too seriously. And anyway, films like these aren’t about realism. To which we who are anal and proud will reply: oh yes, they are! The big idea behind the Bourne films is to make them look realistic. These are spy films that want to break free from that whole post-Bond fantasy action world of Schwarzenegger in True Lies or Vin Diesel in XXX.

The director Paul Greengrass (Bloody Sunday) isn’t interested in showing us exotic locations and erotic babes. Like Bond, Bourne travels a lot (Berlin, Milan and Moscow), but it’s the boring and bland bits of cities that we see. Greengrass doesn’t do big set pieces; he does great small set pieces, as when we see Bourne outmanoeuvre Landy or escape capture. He sends his camera into the heart of the action — where it wobbles like mad — in a technique inspired by gritty shows such as NYPD and reality TV programmes such as the ground-breaking Cops.

This, along with John Powell’s music, gives a real tension to the action sequences and allows us an intimacy with Bourne. That isn’t easy to establish with a guy who has amnesia, and not much in the way of personality, either. It is to Greengrass’s credit that he has taken all the traditional spy stuff of old — the double-crosses, the car chases, shoot-outs, glorious gadgetry — and managed to make them look fresh.

At one time, having Matt Damon as your action hero made about as much sense as casting Woody Allen. And yet it works. He has the face of a thoughtful 14-year-old choirboy. You would never guess he’s a killer — which is perfect cover for a killer. Damon’s Bourne never makes a big deal out of his considerable fighting skills or use of gadgetry.

This sequel has a first-rate group of supporting actors. Allen is perfect as a determined agent dedicated to the truth. In the first film, we had Chris Cooper running around, ordering people in body bags as if they were burgers in sesame buns; here we get Brian Cox as the CIA operative, Ward Abbott, who says his commitment to his job has cost him two marriages and 40 years of “shovelling shit on four continents”.

I must admit that the prospect of seeing this film filled me with dread. Okay, I was wrong. Even people like me, who don’t like spy films, can enjoy this one.

The Bourne Supremacy, 12A, 108 mins, Three stars