Real polynomial interpolation

Question

Let $z_1,\overline{z}_m,\ldots, z_m,\overline{z}_m$ be pairs of complex conugated points in $\mathbb{C}^n$. If considering an interpolation problem of the form $f(z_{i})=a_i$ and $f(\overline{z}_i) =\overline{a}_i$, does there always exist a real solution, i.e. a polynomial $f$ with real coefficients solving the interpolation problem?