quantum field theory - In QFT, why do fermions have to anticommute in order to insure causality?

I have seen this question and I believe I understand the answer to it. However, AFAIK, only for bosons the causality condition is a vanishing commutator. For fermions we expect the anticommutator $[\phi,\phi^\dagger]_+$ to turn zero. The answer given to the question above does not seem to address this.