 Real Grassmann Gaussian integral

For an antisymmetric matrix $A$, a Berezin integral extracts the top coefficient of the exponential. Expansion into pairings gives the Pfaffian, with a constant sign depending on the order of the differentials. This is a polynomial in the entries of $A$, not the positive square root of its determinant: in dimension two it is proportional to $A_{12}$ and changes sign with that entry. The identity $\mathrm{Pf}(A{)}^2=\det A$ determines its square but does not discard its orientation.