The other "problem" that anti-neutrinos have is that they can't just annihilate with ordinary neutrinos. This is because the only particle that could be produced by such an annihilation - the Z0 boson - is heavy, almost invariably much heavier than the COM energy of the neutrino/anti-neutrino pair. This is in stark contrast with "ordinary" [baryonic and charged leptonic] matter and antimatter, which have electric charges and can therefore annihiliate easily into massless photons.
A virtual Z0 could, once in about 4 x 10x blue moons, extremely fleetingly pop into existence from a neutrino/anti-neutrino collision, but the only decay channel then open to it (unless the available energy were at least 2me) would be ... to decay back into a neutrino and anti-neutrino.
As a result, many anti-neutrinos, including those from the cosmic neutrino background, exist and pass through us regularly every day, and we don't even notice them!