  We continue to study the unbounded $\mathfrak{sl}_3$-laminations [IK22], with
a focus on their structures at punctures. A key ingredient is their relation to
the root data of $\mathfrak{sl}_3$. After giving a classification of signed
$\mathfrak{sl}_3$-webs around a puncture, we describe the tropicalization of
the Goncharov--Shen's Weyl group action in detail. We also clarify the
relationship with several other approaches by Shen--Sun--Weng [SSW23] and
Fraser--Pylyavskyy [FP21]. Finally, we discuss a formulation of unbounded
$\mathfrak{g}$-laminations for a general semisimple Lie algebra $\mathfrak{g}$
in brief.
