Consider an irreducible component $D_{i}$ of $\mathscr{X}_{k}$ and write $\Delta_{D_{i}}=\sum_{j\neq i}D_{j}\cap D_{i}$. By adjunction, the pair $(D_{i},\Delta_{D_{i}})$ is log Calabi–Yau, i.e. $D_{i}$ is a smooth projective variety over $k$ and $\Delta_{D_{i}}$ is a divisor such that $K_{D_{i}}+\Delta_{D_{i}}$ is trivial. By [EM21, Theorem 6.14] there exists a Lagrangian torus fibration

$$\phi:\mathcal{U}\rightarrow B\subseteq\Star(v_{D_{i}})\setminus W$$

where $\mathcal{U}$ is a symplectic tubular neighborhood of the $1$-dimensional strata of $\Delta_{D_{i}}$, $B$ is a retract of $\Star(v_{D_{i}})\setminus W$, and $W$ is the union of cells of codimension $\geqslant 2$ in $\Sk(\mathscr{X})$. The fibration $\phi$ is constructed gluing toric moment maps defined in the neighborhood of each stratum curve of $\Delta_{D_{i}}$. Evans and Mauri compare the monodromy $T_{\phi}$ induced by $\phi$ on $B$ to the monodromy $T_{\rho_{\mathscr{X}}}$ induced by the affinoid torus fibration

$${\rho_{\mathscr{X}}}:\rho_{\mathscr{X}}^{-1}(\Star(v_{D_{i}})\setminus W)\rightarrow\Star(v_{D_{i}})\setminus W$$

and conclude that they are dual. This means that given a loop $\gamma\in\pi_{1}(B)\simeq\pi_{1}(\Star(v_{D_{i}})\setminus W)$, we have $T_{\rho_{\mathscr{X}}}(\gamma)=(T_{\phi}(\gamma)^{-1})^{T}$. Thus the affine structure constructed in [NXY19] has a symplectic topological analog. The duality is due to the fact that the image of the moment maps is $M_{\mathbb{R}}$, while the image of the tropicalization map $\val$ is in $N_{\mathbb{R}}$.