  For a finite-type surface $\mathfrak{S}$, we study a preferred basis for the
commutative algebra
$\mathbb{C}[\mathscr{R}_{\mathrm{SL}_3(\mathbb{C})}(\mathfrak{S})]$ of regular
functions on the $\mathrm{SL}_3(\mathbb{C})$-character variety, introduced by
Sikora-Westbury. These basis elements come from the trace functions associated
to certain tri-valent graphs embedded in the surface $\mathfrak{S}$. We show
that 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 character
variety.
