= Reality obstruction for a complex Euclidean gauge condition
For real $A_\mu$ in positive-definite Euclidean signature, $\partial\cdot A$ is real and $A_\mu A_\mu\ge0$. Therefore $\partial\cdot A+iA^2=0$ 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.
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