 Reality obstruction for a complex Euclidean gauge condition

For real $A_{\mu }$ in positive-definite Euclidean signature, $\partial \cdot A$ is real and $A_{\mu }{A}_{\mu }\ge 0$. Therefore $\partial \cdot A+i{A}^2=0$ forces $A^2=0$, and hence $A=0$. Its only real gauge orbit is the zero-field orbit; a field with nonzero electromagnetic curvature cannot be gauge-transformed to zero. Consequently this complex equation is not an admissible gauge fixing for general Hermitian Euclidean U(1) gauge symmetry configurations. A formal complexified treatment needs a specified contour and cannot be justified by a real delta-functional insertion.