If $S$ is a K3 surface of Type III, and $\mathscr{X}/R$ a minimal model of $S$, then the map $\rho_{\mathscr{X}}$ is an affinoid torus fibration away from the vertices of $\mathcal{D}(\mathscr{X}_{k})$.
The induced $\mathbb{Z}$-affine structure with isolated singularities on the 2-sphere matches the one constructed in [GHK15, 1.2] and [Eng18, Proposition 3.10], and has no singularity at a vertex $v_{D}$ if and only if the corresponding component $D$ of $\mathscr{X}_{k}$ is toric.