  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.
