We will now show that the affine structure we constructed on $\Sk(X)\setminus\Gamma$ is semi-simple polytopal in the sense of [RZ21a, Definition 4]. Integral affine manifolds with semi-simple polytopal singularities are the tropical analog of local complete intersections in algebraic geometry, and are the relevant class of affine structures on the base of the topological SYZ fibration in the context of the Gross–Siebert program. Indeed, given such a manifold $B$, Ruddat and Zharkov construct a topological space $Y$ and torus fibration $Y\rightarrow B$ with discriminant of codimension 2 in $B$, inducing the given affine structure. In [RZ21a] the authors describe the strategy in the 3-dimensional case; the general results will appear in [RZ], building on the local constructions of [RZ21b].
In the case of the quintic 3-fold, let $p=p_{ijk}$ be a vertex of the discriminant contained in the interior of a 2-face $\tau$, and $q=p_{ik}$ a vertex contained in the interior of an edge $e$ of $\tau$. Up to relabelling, we may assume that the lattice $L_{p}$ of invariant vectors around $p$ (i.e. the sections of the sheaf of integral affine tangent vectors on a small neighbourhood of $p$) is freely generated by $v_{i}$ and $v_{j}$, in which case the three monodromy matrices around $p$ are of the form $T=\Id+5v_{k}^{\vee}\otimes w$ for some primitive $w\in L_{p}$. Hence, writing $L(p)=L_{p}$ and $L^{\vee}(p)=5L_{p}^{\bot}$, as well as $L(q)=L_{q}$ and $L^{\vee}(q)=5L_{q}^{\bot}$ we see that we are in the setting of [RZ21a]: the singularities of the affine structure are semi-simple abelian. Moreover, the vertices $p_{ijk}$ are negative, while the vertices $p_{ik}$ are positive.
Note that $\tau$ can be canonically realized inside $L_{p}$, sending the vertex $v_{k}$ to the origin; in addition we set $\tau^{\vee}=<0,5v_{k}^{\vee}>$ to be the convex hull of $0$ and $5v_{k}^{\vee}$ in $\subset L^{\vee}(p)$. The three loops described above are canonically indexed by the edges of $\tau$, and hence by the pairs $(e,f)$, with $e$ an edge of $\tau$ and $f$ the edge of $\tau^{\vee}$. The upshot of working with $5L_{p}^{\bot}$ instead of $L_{p}^{\bot}$ (and similarly for $q$) is now that the monodromy along the loop $\gamma_{e,f}$ is now simply given by the formula $T(\gamma_{e,f})=\Id+e\otimes f$.
Similarly for $q$, we realize the edge $e_{ik}$ inside $L_{q}$ as the unit segment, and set $e^{\vee}=<0,5v_{i}^{\vee},5v_{k}^{\vee}>\subset L^{\vee}(q)$. Then we may once again label the three loops around $q$ by pairs $(e,f)$ with $e=e_{ik}$ and $f$ an edge of $e^{\vee}$, so that the formula $T(\gamma_{e,f})=\Id+e\otimes f$ holds.
Since $e_{ik}$ is a face of $\tau$, we conclude from this that our affine structure is semi-simple polytopal.

In [Rua01], Ruan develops a symplectic method based on gradient flow and constructs a Lagrangian torus fibration for Fermat type quintic Calabi–Yau hypersurfaces

later extended to generic quintic hypersurfaces in toric varieties. The idea is to realize $\Sk(\mathscr{X})$ very explicitely as the boundary of the standard 4-simplex $\tau^{4}=\{\sum_{i=0}^{4}w_{i}=1\}\subset\mathbb{R}_{\geqslant 0}^{5}$, and to spread the map:

to the nearby fibers using a gradient flow. This yields a Lagrangian fibration on the $\mathscr{X}_{t}$’s for small enough $t$, which Ruan expects to be deformable towards a special Lagrangian fibration.
In addition, he describes the discriminant locus and the monodromy transformations of the expected special Lagrangian fibration, assuming that the singular locus is of codimension $2$. The predictions in [Rua01, §4.4, §4.5] match precisely our computations above.

In [Gro01] Gross defines a class of topological $3$-dimensional torus fibrations and proves they admit dual fibration. Building on Ruan’s description of monodromy, Gross shows that generic quintic threefolds in $\mathbb{P}^{4}$ can be endowed with such a fibration. It follows that the induced integral affine structure on the sphere $\mathbb{S}^{3}$ coincides with the one in Section 4.7.

Note that both in Ruan’s and in Gross’ aforementioned works, the polyhedral decomposition on $\mathbb{S}^{3}$ is induced by the intersection complex of the central fiber $\mathscr{X}_{0}$ (i.e., vertices correspond to zero-dimensional strata of the special fiber and so on); we work instead with the dual intersection complex associated with $\mathscr{X}_{0}$, which is isomorphic to the intersection complex in the examples we are considering.