One version of Hartogs' extension theorem is the following (see, e.g. [1], Theorem 5B, p. 50).
Theorem. Let $U \subset \mathbb{C}^n$ be open and let $X \subset U$ be a complex-analytic subvariety of codimension $>1$. Then, any holomorphic function $U \setminus X \to \mathbb{C}$ has a unique holomorphic extension $U \to \mathbb{C}$.
Question. Is this theorem still true if we replace $X$ by a real-analytic subvariety of codimension $>2$?
Note that I am still considering holomorphic functions. (The real-analytic version of Hartogs' theorem is false.)
References.
[1] Whitney, Hassler. Complex analytic varieties. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. xii+399 pp.