Non-uniqueness of characteristic curves of folding rational maps

A folding rational map is a postcritically finite rational map $f$ admitting an essential Jordan curve $\beta$, disjoint from the postcritical set $P_f$, whose preimage is disconnected and consists of Jordan curves homotopic to $\beta$ relative to $P_f$. Such $\beta$ is called a characteristic curve of $f$. We construct a real postcritically finite rational map $f$ of degree $6$ with $\mathrm{\#}{P}_f=4$, which admits two non-homotopic characteristic curves $\beta$ and ${\beta }^{\mathrm{\prime }}$. It is hyperbolic with orbifold signature $(2,2,3,\mathrm{\infty })$. This non-uniqueness is not a Lattès phenomenon, nor does it arise from a mating decomposition. The map is constructed piecewise, and a numerical model is provided to illustrate its dynamics. Thurston obstructions are excluded by arc-lifting and intersection-number arguments.