  We introduce a topological intersection number for an ordered pair of
$\operatorname{SL}_3$-webs on a decorated surface. Using this intersection
pairing between reduced $(\operatorname{SL}_3,\mathcal{A})$-webs and a
collection of $(\operatorname{SL}_3,\mathcal{X})$-webs associated with the
Fock--Goncharov cluster coordinates, we provide a natural combinatorial
interpretation of the bijection from the set of reduced
$(\operatorname{SL}_3,\mathcal{A})$-webs to the tropical set
$\mathcal{A}^+_{\operatorname{PGL}_3,\hat{S}}(\mathbb{Z}^t)$, as established by
Douglas and Sun in \cite{DS20a, DS20b}. We provide a new proof of the flip
equivariance of the above bijection, which is crucial for proving the
Fock--Goncharov duality conjecture of higher Teichm\"uller spaces for
$\operatorname{SL}_3$.
