Thanks to reading this thread, I asked the name of the vaccine used by my new vet when Fred and Vic were re-vaccinated (after a two year break), when she looked puzzled, I showed her the bit of paper upon which I'd scribbled "Fevaxyn Pentofel" and she had an "Ah, now I understand" looked. She explained that this was a 5 dose vaccine, containing the two flu strains, enteritis, leukemia and Chlamydia.
Apparently, it's the Chlamydia element which causes the problems: this isn't routinely vaccinated against because the vaccine causes more problems than the illness in cats. Usually, the illness is a mild eye watering which is harmless, apparently.