Let $X/K$ be a maximally degenerate $K3$ surface and let $\mathscr{X}/R$ be a minimal model of $X$ with reduced special fiber $\mathscr{X}_{k}=\sum_{i\in I}D_{i}$. The dual complex $\mathcal{D}(\mathscr{X}_{k})$ is well-known to be a triangulated sphere, whose vertices correspond to the irreducible components of $\mathscr{X}_{k}$.
We focus our attention to such a vertex $v_{D}$, and hence to the corresponding irreducible component $D$ of $\mathscr{X}_{k}$, which has boundary $\Delta_{D}\coloneqq\sum_{i=1}^{r}(D_{i}\cap D)=\sum_{i=1}^{r}C_{i}$. Since the simple normal crossing curve $\Delta_{D}\in\lvert-K_{D}\rvert$ is an anticanonical curve by adjunction, it follows from general surface theory that $\Delta_{D}$ is a cycle of rational curves $(C_{i})_{i\leq r}$, whose geometry is encoded by the $b_{i}=-(C_{i}\cdot D)=-(C_{i}^{2})_{D}$. We label the curves so that for $i\leq r$, $C_{i}\cap C_{i+1}\neq\varnothing$, with convention $C_{r+1}=C_{1}$.

One can associate to the pair $(D,\Delta_{D})$ a pseudo-fan, which is a singular affine structure on $\mathbb{R}^{2}$, singular at most at $0$. The singularity at $0$ is a way to measure the defect of $(D,\Delta_{D})$ of being toric: the affine structure affine extends smoothly at $0$ if and only $(D,\Delta_{D})$ is a toric pair [Eng18, Proposition 3.9].
The construction, as explained in [GHK15, §1.2], is the following. For each node $p_{i}=C_{i}\cap C_{i+1}$, consider a cone $\sigma_{i}\coloneqq\mathbb{R}_{\geq 0}v_{i}+\mathbb{R}_{\geq 0}v_{i+1}\subset\mathbb{R}^{2}$, $(v_{i},v_{i+1})$ being a basis of the lattice $\mathbb{Z}^{2}$. The cones $\sigma_{i}$ and $\sigma_{i+1}$ are then glued to each other along $\mathbb{R}_{\geq 0}v_{i+1}$, and the affine structure is extended through the edge by pretending that the pair $(D,\Delta_{D})$ is toric. If the pair was toric, the $\sigma_{i}$’s would be the maximal cones of its fan, and the relation

would hold by Eq. 1.2.4, so that the chart $\psi_{i}:\sigma_{i}\cup\sigma_{i+1}$ that defines the $\mathbb{Z}$-affine structure satisfies $\psi_{i}(0)=0$, $\psi_{i}(v_{i})=(1,0)$, $\psi_{i}(v_{i+1})=(0,1)$ and $\psi_{i}(v_{i+2})=(-1,b_{i+1})$, and is extended by dilatation. The unions of the $\sigma_{i}$’s glued along the successive edge is homeomorphic to $\mathbb{R}^{2}$, and we obtain this way an $\mathbb{Z}$-affine structure away from the origin, extending to $0$ if and only the pair is toric.

It follows from Proposition 3.1.1 that the singular $\mathbb{Z}$-affine structure induced by the Berkovich retraction $\rho_{\mathscr{X}}$ coincides with the one described above. We now determine the monodromy around the singularities.