I'm actually wrong about that -- Tanimoto does return 0.0 when the intersection is empty (but, the union is not) which is hewing to the mathematically-correct definition. I could argue myself either way on the behavior but figure best to leave it as it is. This isn't a typically-used metric anyhow.
But for purposes of the discussion here it doesn't change the analysis. A bunch of 0s do the same thing as NaNs, and in fact make the scenario I describe a little more possible.
