My understanding is that the anticommutator relationship shows that the many-particle wave function is asymmetric relative to particle exchange, which is related to the Pauli exclusion principle and also half-integer spin, which, in turn, defines Fermi-Dirac statistic, where the maximal value can't exceed one, unlike in the case of the Bose-Einstein statistics.
The spin statistic theorem talks about this, you can find it here.
To answer your question, they are compatible in the sense that two particles can't occupy the same quantum state.