We consider the blow-up $h_{ijkl}:\mathscr{W}_{ijkl}\rightarrow\mathscr{X}_{ijkl}$ of the surfaces $S_{mm^{\prime}}$ in lexicographical order with respect to $(i,j,k,l,h)$; we denote by $E_{mm^{\prime}}$ the corresponding exceptional divisors. The skeleton $\Sk(\mathscr{W}_{ijkl})$ consists of the union of the skeleton $\Sk(X)$ with four additional $3$-cells for each 2-dimensional face $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$ of $\Sk(X)$: for each ordered triple $m<m^{\prime}<m^{\prime\prime}$, the union of the additional cells is isomorphic to $\Sk(\mathscr{V}_{123})$ in Section 4.3, where we identify $v_{1}=v_{m}$, $v_{2}=v_{m^{\prime}}$ and $v_{3}=v_{m^{\prime\prime}}$. The retraction $\rho_{\mathscr{X}_{ijkl}}$ collapses the additional faces onto $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$ as $\rho_{\mathscr{U}_{12}}\circ\rho_{\mathscr{V}_{12}}$.

Given another order $(i^{\prime},j^{\prime},k^{\prime},l^{\prime},h^{\prime})$ on $\{1,2,3,4,5\}$, the skeleton $\Sk(\mathscr{W}_{i^{\prime}j^{\prime}k^{\prime}l^{\prime}})$ coincides with $\Sk(\mathscr{W}_{ijkl})$ as subspace of $X^{\an}$; we denote this simply by $\Sk(\mathscr{W})$. Instead, the triangulation and the retraction depend on the order. Our goal is therefore to construct a model $\mathscr{Z}$ which dominates all models $\mathscr{W}_{ijkl}$ regardless of the order, so that all retractions $\rho_{\mathscr{W}_{ijkl}}$ factors through $\rho_{\mathscr{Z}}$.

Along the same lines of Section 4.3, we define $\mathscr{Z}$ as the blow-up of $\mathscr{W}_{ijkl}$ along $D_{m^{\prime}}\cap E_{mm^{\prime\prime}}$ and $E_{mm^{\prime}}\cap D_{m^{\prime\prime}}^{\prime}$, for all ordered triples $m<m^{\prime}<m^{\prime\prime}$ in the order $(i,j,k,l,h)$:

$\begin{array}[]{ccccccccc}\mathscr{Z}&\xrightarrow[\text{for all }m<m^{\prime}<m^{\prime\prime}]{\begin{subarray}{c}\text{blow-up of }\\ D_{m^{\prime}}\cap E_{mm^{\prime\prime}},E_{mm^{\prime}}\cap D_{m^{\prime\prime}}^{\prime}\end{subarray}}&\mathscr{W}_{ijkl}&\xrightarrow[h_{ijkl}]{\begin{subarray}{c}\text{blow-up of }S_{mm^{\prime}}\\ \text{for all }m<m^{\prime}\end{subarray}}&\mathscr{X}_{ijkl}&\xrightarrow[g_{ijkl}]{\begin{subarray}{c}\text{blow-up of }\\ D_{i},D_{j},D_{k},D_{l}\end{subarray}}&\mathscr{X}.\\ \cup&&\cup&&\cup&&\\ E_{mm^{\prime}m^{\prime\prime}},E^{\prime}_{mm^{\prime}m^{\prime\prime}}&&E_{mm^{\prime}}&&S_{mm^{\prime}}&&\end{array}$

We denote by $E_{mm^{\prime}m^{\prime\prime}}$ and $E^{\prime}_{mm^{\prime}m^{\prime\prime}}$ the corresponding exceptional divisors, and deduce from the local study of these morphisms in Section 4.3 that $\Sk(\mathscr{Z})$ is obtained from $\Sk(\mathscr{W})$ by adding a new $3$-cell $\tau_{mm^{\prime}m^{\prime\prime}}:=<v_{mm^{\prime}},v_{mm^{\prime\prime}},v_{m^{\prime}m^{\prime\prime}},v^{\prime}_{mm^{\prime}m^{\prime\prime}}>$ for each triple $m<m^{\prime}<m^{\prime\prime}$.

We now define the combinatorial retraction of $\Sk(\mathscr{Z})$ onto $\Sk(X)$: given the 2-cell $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$, we identify $v_{1}=v_{m}$, $v_{2}=v_{m^{\prime}}$ and $v_{3}=v_{m^{\prime\prime}}$ and contract onto $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$ the additional cells of $\Sk(\mathscr{Z})$ over $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$, via the combinatorial retraction $\pi^{\prime}=\rho\circ\kappa$ constructed in Section 4.4. With a slight abuse of notation, we still denote this map by $\pi^{\prime}$. By construction, the composition $\pi=\pi^{\prime}\circ\rho_{\mathscr{Z}}$

$\begin{array}[]{ccccccc}&\pi:&X^{\an}&\xrightarrow{\rho_{\mathscr{Z}}}&\Sk(\mathscr{Z})&\xrightarrow[\begin{subarray}{c}\text{combinatorial}\\ \text{retraction}\end{subarray}]{\pi^{\prime}}&\Sk(X)\\ &\rotatebox[origin={c}]{270.0}{$=$}&\cup&&\cup&&\cup\\ \text{over }\Star(v_{m})^{\prime}\text{\hskip 10.0pt}&\rho_{\mathscr{X}_{ijkl}}:&\pi^{-1}(\Star(v_{m})^{\prime})&\xrightarrow{\rho_{\mathscr{Z}}}&\Sk(\mathscr{Z})&\xrightarrow{\rho_{\mathscr{X}_{ijkl}}}&\Star(v_{m})^{\prime}\end{array}$

coincides with $\rho_{\mathscr{X}_{ijkl}}$ over $\Star(v_{m})^{\prime}$ for any order $(i,j,k,l,m)$ on $\{1,2,3,4,5\}$. In other words, around each vertex $v_{m}$, the map $\pi$ is the Berkovich retraction induced by a small resolution $\mathscr{X}_{ijkl}$ of $\mathscr{X}$, where the strict transform of $D_{m}$ is isomorphic to $D_{m}$, thus in particular $\mathring{D_{m}}\subset D_{m}$ is a torus embedding.

For each $2$-dimensional face $<v_{m},v_{m^{\prime}},v_{m^{\prime\prime}}>$ of $\Sk(X)$, we denote by $\Gamma_{mm^{\prime}m^{\prime\prime}}$ the graph defined in Definition 3.2.1, and its vertices by $p_{mm^{\prime}},p_{mm^{\prime\prime}},p_{m^{\prime}m^{\prime\prime}}$ and $p_{mm^{\prime}m^{\prime\prime}}$. We set

By construction, around any point of $\Sk(X)\setminus\Gamma$, the retraction $\pi$ is equal to the Berkovich retraction induced by a suitable minimal model $\mathscr{X}_{ijkl}$ of $X$. It follows from the results in [NXY19] that $\pi$ induces an integral affine structure with singularities on $\Sk(X)$. By Theorem B and Corollary C, we obtain that this integral affine structure has no singularities outside $\Gamma$. We will furthermore prove in the next subsection that this affine structure does not extend across any edge of $\Gamma$, i.e. is indeed singular along $\Gamma$.