Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces II: naturality

In a companion paper (arXiv 2011.01768), we constructed nonnegative integer coordinates $\Phi_\mathscr{T}(\mathscr{W}_{3, \hat{S}}) \subset \mathbb{Z}_{\geq 0}^N$ for the collection $\mathscr{W}_{3, \hat{S}}$ of reduced $\mathrm{SL}_3$-webs on a finite-type punctured surface $\hat{S}$, depending on an ideal triangulation $\mathscr{T}$ of $\hat{S}$. We show that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $\mathscr{T}^\prime$ is chosen, then the coordinate change map relating $\Phi_\mathscr{T}(\mathscr{W}_{3, \hat{S}})$ to $\Phi_{\mathscr{T}^\prime}(\mathscr{W}_{3, \hat{S}})$ is a tropical $\mathcal{A}$-coordinate cluster transformation. We can therefore view the webs $\mathscr{W}_{3, \hat{S}}$ as a concrete topological model for the Fock-Goncharov-Shen positive integer tropical points $\mathcal{A}_{\mathrm{PGL}_3, \hat{S}}^+(\mathbb{Z}^t)$.