= Solution
A <Berezin integral> over independent <Grassmann variables> selects the coefficient containing every variable. Fix the real orientation by $\int d\omega_1\cdots d\omega_{2m}\,\omega_{2m}\cdots\omega_1=1$. For an <antisymmetric matrix> $A$, only the order-$m$ term in the exponential survives:
$$
\frac{(-1)^m}{2^m m!}\left(\sum_{i,j}\omega_iA_{ij}\omega_j\right)^m.
$$
Anticommutation collects its top coefficient into the signed sum over pairings that defines the <Pfaffian>. Reversing the order of $2m$ variables gives $(-1)^{m(2m-1)}=(-1)^m$, cancelling the prefactor sign. The <real Grassmann Gaussian integral> is therefore
$$
\boxed{\int d\omega_1\cdots d\omega_{2m}\,e^{-\omega^TA\omega/2}=\operatorname{Pf}(A)}
$$
with this orientation, and a fixed sign times this answer with other orientations. In dimension two, write $A_{12}=a$; the exponential is $1-a\omega_1\omega_2=1+a\omega_2\omega_1$, and integration gives $a$. Although $\operatorname{Pf}(A)^2=\det A$, writing a positive square root $\sqrt{\det A}$ would give $|a|$ in this test and would lose the correct sign.
For the complex <Grassmann Gaussian integral>, treat $\theta_a$ and $\bar\theta_a$ as independent integration variables. Every surviving top term contains each $\theta$ and each $\bar\theta$ once. Expanding the exponential assigns one matrix entry from every row and column, and anticommutation supplies the permutation sign. Hence
$$
\boxed{\int\prod_{a=1}^n d\theta_a\prod_{a=1}^n d\bar\theta_a\,e^{-\bar\theta^TM\theta}=C_n\det M,}
$$
where $C_n$ is a nonzero constant fixed by the order and normalization of the differentials, independent of $M$. This argument works for singular matrices as well and does not assume that $M$ is Hermitian.
For the explicit $n=2$ check, put $Q=\bar\theta_aM_{ab}\theta_b$. Moving all the $\theta$ variables before the barred ones gives
$$
\frac12Q^2=(-M_{11}M_{22}+M_{12}M_{21})\theta_1\theta_2\bar\theta_1\bar\theta_2.
$$
The linear and constant terms integrate to zero. Thus the integral is a fixed orientation sign times $M_{11}M_{22}-M_{12}M_{21}$, exactly the <determinant>. Choosing the top-monomial integral to be $-1$ gives $C_2=1$.
For the required <real-complex compatibility of Grassmann Gaussian integrals>, let $u_a=\omega_a$ and $v_a=\omega_{n+a}$. Substitution gives
$$
\bar\theta^TM\theta=\frac12\left(u^TMu+v^TMv+i u^TMv-i v^TMu\right)=\frac12(u^TMu+v^TMv).
$$
Indeed $v^TMu=-u^TM^Tv=u^TMv$ because the variables anticommute and $M^T=-M$. The real quadratic matrix is consequently $A=\operatorname{diag}(M,M)$, of size $2n$. The linear change of <Grassmann variables> has a constant nonzero <Berezin integration> Jacobian, so it changes only the allowed normalization factor. For even $n$, the two blocks give
$$
\boxed{\operatorname{Pf}(A)=\operatorname{Pf}(M)^2=\det M.}
$$
For odd $n$, $\det M=\det(-M)=(-1)^n\det M$ implies $\det M=0$. Each of the separate $u$ and $v$ quadratic exponentials has only even degree and cannot saturate an odd number of variables, so the real integral vanishes too. This covers the odd case without attempting to define a <Pfaffian> of an odd-dimensional matrix.
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