The multi-valued quasimöbius mappings on the Riemann sphere

Suppose that a multi-valued mapping $F: D\to 2^{\overline{\Bbb C}}$ of a domain $D$ in the sphere $\overline{\Bbb C}$ with disjoint images of distinct points boundedly distorts the Ptolemaic characteristic of generalized tetrads (quadruples of disjoint compact sets). Suppose that the image $F(x)$ of each $x\in D$ has at most $N$ components, each of which is a continuum of bounded turning. Then $F$, up to the values at some isolated branch points, is the inverse of a mapping with bounded distortion in the sense of Reshetnyak. In particular, if $D= \overline{\Bbb C}$ then the left inverse to $F$ is the composition of a quasiconformal automorphism of $\overline{\Bbb C}$ and a rational function.