>Is there some mechanism I haven't run across that produces extra Tritium or extra neutrons somehow?

Yes. In addition to breeding tritium from (n,t) reactions on 6Li, you can multiply neutrons through, e.g., n + 9Be -> 2*4He + 2 n, and can also breed tritium without losing a neutron through (n,t) reactions on Li7. Both can, in principle, lead to tritium production systems capable of replacing the fuel at the rate at which it's consumed.

Such systems haven't been demonstrated at scale, however, and would have to be co-located with the power plant, leading to all sorts of complications (not to mention NIMBY concerns) but there is nothing fundamental that is stopping such a recovery system from being designed.