Like all plant rescue diagnoses, it is difficult to answer without having seen it. However, your description matches a fungal infection called Puccinia allii, an onion rust which can occur on different varieties of onion. The fungus requires high humidity to spread, which is why it occurs less often in well ventilated garden plants. The fungus also survives the winter on the remains of affected plants.
So what can you do? Thoroughly remove any affected foliage. Harvest the chives on a regular basis, so that the fungus has less time to spread out. Fertilise with nitrogen only sparingly, as excessive nitrogen causes the development of soft plant tissue which is receptive to fungus.
If the location has been infected for some years, as you suggest, you can either use a fungicide for fungal infections on herbs and vegetables or rebuild your stock of chives and wild garlic somewhere else with clean propagation material.