  Generalizing the work of Fock--Goncharov on rational unbounded laminations,
we give a geometric model of the tropical points of the cluster variety
$\mathcal{X}_{\mathfrak{sl}_3,\Sigma}$, which we call unbounded
$\mathfrak{sl}_3$-laminations, based on the Kuperberg's $\mathfrak{sl}_3$-webs.
We introduce their tropical cluster coordinates as an
$\mathfrak{sl}_3$-analogue of the Thurston's shear coordinates associated with
any ideal triangulation. As a tropical analogue of gluing morphisms among the
moduli spaces $\mathcal{P}_{PGL_3,\Sigma}$ of Goncharov--Shen, we describe a
geometric gluing procedure of unbounded $\mathfrak{sl}_3$-laminations with
pinnings via ``shearings''. We also investigate a relation to the graphical
basis of the $\mathfrak{sl}_3$-skein algebra [IY23], which conjecturally leads
to a quantum duality map.
