I'd have thought it's very unlikely to be the coupling (at least in an unrepairable sense). The brakes work normally because the trailer tries to overtake the car when the car is slowing down. That pushes in the piston on the hitch, and activates the brakes.
However, reversing the car also pushes on that piston, and would normally activate the brakes (and it sounds as though that's the case for you here Clive). As Buttermilk says, there are two types of mechanism to prevent this from happening - manual and automatic. The first thing to do would be to find out if it's manual or automatic, and then go from there.
Another thing you can do - If it's a four-wheel trailer, get a ramp or a piece of wood, then drive up onto it with the "other" wheel so that each wheel in turn is lifted off the ground. Then rotate the now airborne wheel forwards and backwards by hand. If it will rotate freely both ways when the hitch piston is in the "out" position, next try compressing the hitch piston using a rope or strap and see what happens. The wheel should now brake forward rotation, but then the brake should unlock to allow backward rotation.
Perhaps try that and report back?