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The "real" integral evalautes to the Pfaffian of $A$ where for an $2n$-by$2n$ skew symmetric matrix $A$ $$ {\rm Pf}A= \frac 1{2^n n!} \epsilon_{i_1, \ldots, i_{2n}} A_{i_1,i_2}\ldots A_{2n-1,2n}. $$
The Pfaffian has the property that $({\rm Pf} A)^2= {\rm det}A$, and has applications many places in combinatorics. For Majorana fermion path integrals it replaces the one-loop Matthews-Salam determinant.