Yes, of course, after we shrink $U$. A convergent Taylor series at a real point converges in some complex neighborhood of this point.
Added reply to your comment: you can apply identity theorem. Two real analytic functions coinciding on an open set of $R^n$ coincide in a complex neighborhood of this set.
Yes, of course, after we shrink $U$. A convergent Taylor series at a real point converges in some complex neighborhood of this point.