Real-complex compatibility of Grassmann Gaussian integrals ($\mathrm{Pf}(\mathrm{diag}(M,M))=\det M$)

For an even-dimensional antisymmetric matrix $M$, put $\theta =(u+iv)/\sqrt{2}$ and $\bar{\theta }=(u-iv)/\sqrt{2}$, with independent Grassmann variables $u,v$. Anticommutation cancels the mixed terms, giving ${\bar{\theta }}^{T}M\theta =(u^{T}Mu+v^{T}Mv)/2$. The constant Grassmann change-of-variables formula changes only the normalization. The two real Pfaffians multiply to $\mathrm{Pf}(M{)}^2=\det M$, agreeing with the complex Grassmann Gaussian integral. In odd dimension $\det M=0$ and both top-degree integrals vanish.