Tropical Fock-Goncharov coordinates for ${\mathrm{S}\mathrm{L}}_3$-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$forthecollection$\mathscr{W}_{3, \hat{S}}$ofreduced$$\mathrm{S}\mathrm{L}$_3$-websonafinite-typepuncturedsurface$\hat{S}$, depending on an ideal triangulation $\mathcal{T}$ of $\widehat{S}$. We show that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation ${\mathcal{T}}^{\mathrm{\prime }}$ is chosen, then the coordinate change map relating ${\mathrm{\Phi }}_{\mathcal{T}}({\mathcal{W}}_{3,\widehat{S}})$ to ${\mathrm{\Phi }}_{{\mathcal{T}}^{\mathrm{\prime }}}({\mathcal{W}}_{3,\widehat{S}})$ is a tropical $\mathcal{A}$-coordinate cluster transformation. We can therefore view the webs ${\mathcal{W}}_{3,\widehat{S}}$ as a concrete topological model for the Fock-Goncharov-Shen positive integer tropical points ${\mathcal{A}}_{{\mathrm{P}\mathrm{G}\mathrm{L}}_3,\widehat{S}}^{+}({\mathbb{Z}}^t)$.