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

In a companion paper (arXiv 2011.01768) we constructed non-negative integer coordinates $Phi_mathcal{T}$ for a distinguished collection $mathcal{W}_{3, widehat{S}}$ of $mathrm{SL}_3$-webs on a finite-type punctured surface $widehat{S}$, depending on an ideal triangulation $mathcal{T}$ of $widehat{S}$. We prove that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $mathcal{T}^prime$ is chosen then the coordinate change map relating $Phi_mathcal{T}$ and $Phi_{mathcal{T}^prime}$ is a prescribed tropical cluster transformation. Moreover, when $widehat{S}=Box$ is an ideal square, we provide a topological geometric description of the Hilbert basis (in the sense of linear programming) of the non-negative integer cone $Phi_mathcal{T}(mathcal{W}_{3, Box}) subset mathbb{Z}_{geq 0}^{12}$, and we prove that this cone canonically decomposes into 42 sectors corresponding topologically to 42 families of $mathrm{SL}_3$-webs in the square.