Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic? I can deduce this statement from Remmert-Stein theorem when $\dim_R S < 2 \dim_R Z$, where $S$ is the singular set of $Z$. Also I can deduce this statement from Skoda-El Mir theorem when $S$ is pluripolar. I suspect that it should be true in bigger generality, maybe always.