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For a finite-type surface $\mathfrak{S}$, we study a preferred basis for thecommutative algebra$\mathbb{C}[\mathscr{R}_{\mathrm{SL}_3(\mathbb{C})}(\mathfrak{S})]$ of regularfunctions on the $\mathrm{SL}_3(\mathbb{C})$-character variety, introduced bySikora-Westbury. These basis elements come from the trace functions associatedto certain tri-valent graphs embedded in the surface $\mathfrak{S}$. We showthat this basis can be naturally indexed by non-negative integer coordinates,defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions.These coordinates are related, by the geometric theory of Fock and Goncharov,to the tropical points at infinity of the dual version of the charactervariety.

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