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

$$\mathscr{X}=\{z_{0}\ldots z_{4}+t(z_{0}^{4}+\ldots+z_{4}^{4})=0\}\subset\mathbb{P}^{4}_{\mathbb{C}}\times\mathbb{D},$$

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:

$$F:\mathscr{X}_{0}\longrightarrow\partial\tau^{4}$$

$$[z_{0}:\ldots:z_{4}]\longmapsto\Bigg(\frac{\lvert z_{0}\rvert^{2}}{\lVert z\rVert^{2}},\ldots,\frac{\lvert z_{4}\rvert^{2}}{\lVert z\rVert^{2}}\Bigg)$$

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.