  Let $f$ be a harmonic map from a Riemann surface to a Riemannian
$n$-manifold. We prove that if there is a holomorphic diffeomorphism $h$
between open subsets of the surface such that $f\circ h = f$, then $f$ factors
through a holomorphic map onto another Riemann surface. If such $h$ is
anti-holomorphic, we obtain an analogous statement. For minimal maps, this
result is well known and is a consequence of the theory of branched immersions
of surfaces due to Gulliver-Osserman-Royden. Our proof relies on various
geometric properties of the Hopf differential.
