  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
$\mathcal{A}_{\mathrm{PGL}_3, \hat{S}}^+(\mathbb{Z}^t)$.
