= Solution
Write $F[A]=\partial_\mu A_\mu+iA_\mu A_\mu$ and choose the <U(1) gauge symmetry> convention $A_\mu\mapsto A_\mu+\partial_\mu\alpha$. Its infinitesimal variation is
$$
\delta_\alpha F[A]=M_A\alpha,\qquad M_A=\partial^2+2iA_\mu\partial_\mu.
$$
A formal <BRST symmetry> construction uses $sA_\mu=\partial_\mu c$, $sc=0$, $s\bar c=ih$, $sh=0$ and the <gauge-fixing fermion> $\Psi=\int\bar c(F-i\xi h/2)\,d^4x$. Acting with the odd differential gives
$$
\boxed{S_{\mathrm{gf+gh}}=s\Psi=\int d^4x\left[ihF[A]+\frac\xi2h^2+\bar c\,\mathcal O_Ac\right],\qquad\mathcal O_A=-\partial^2-2iA\cdot\partial.}
$$
For a real admissible gauge functional, integration over $h$ gives $F^2/(2\xi)$ in the <Euclidean action>, and $\xi\to0$ imposes the gauge condition. Overall constant phases in the <Faddeev-Popov determinant> can be absorbed in the measure convention. Here <nonlinear Abelian gauge fixing> makes $\mathcal O_A$ depend on $A$. The action for the <Faddeev-Popov ghost fields> contains the interaction $-2i\bar cA_\mu\partial_\mu c$, so the <Faddeev-Popov ghost fields> cannot be discarded as in a linear Abelian gauge, even though the $U(1)$ <Adjoint representation> is trivial.
There is a genuine <reality obstruction for a complex Euclidean gauge condition> in the source. For a Hermitian $U(1)$ <gauge field>, each Euclidean component $A_\mu$ is real. The real and imaginary parts of $F=0$ separately require $\partial\cdot A=0$ and $\sum_\mu A_\mu^2=0$, hence $A=0$. For example $A_2=Bx_1$, with other components zero and $B\ne0$, has curvature $F_{12}=B$; no <gauge transformation> can put it on this slice because curvature is gauge invariant. Thus the stated condition is not a <gauge fixing> of general real Euclidean configurations. The boxed action is the intended formal complex-gauge construction. A legitimate complexified contour prescription would be additional data, not an ordinary real delta-functional enforcing this printed condition.
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