Covariance of Grassmann bilinears ($⟨(\bar{\theta }M\theta )(\bar{\theta }N\theta ){⟩}_c=-\mathrm{tr}(MCNC)$)

For $C=B^{-1}$, ordered Grassmann Gaussian contractions give $⟨\bar{\theta }M\theta ⟩=-\mathrm{tr}(MC)$ and $⟨(\bar{\theta }M\theta )(\bar{\theta }N\theta )⟩=\mathrm{tr}(MC)\mathrm{tr}(NC)-\mathrm{tr}(MCNC)$. The disconnected contribution has two fermion loops; the connected contribution has one, giving its minus sign. This covariance is an algebraic normalized Berezin integral, rather than a positivity statement about a probability measure.