If $f(a+re^{it})\in \Bbb{R}$ for all $t\in \Bbb{R}$ then $f$ is constant.
Solution 1:
I think the shortest solution is to use the maximum principle for harmonic functions. If $f = u + iv$ (where $u$ and $v$ are real-valued), your assumption shows that $v = 0$ on the circle $\partial D(a,r)$.
By the maximum princple (applied on $v$ and $-v$), it follows that $v = 0$ on $D(a,r)$, and by the identity principle thus on $U$. In other words, $f$ must be real-valued on $U$ and this in turns (for example via Cauchy-Riemann) implies that $f$ is constant.