  Let $\boldsymbol{\Sigma}:=(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with
marked points $\mathbb{M}\subset\partial\Sigma\neq\varnothing$ on the boundary,
and punctures $\mathbb{P}\subset\Sigma\setminus\partial\Sigma$, and $T$ an
arbitrary tagged triangulation of $\boldsymbol{\Sigma}$ in the sense of
Fomin-Shapiro-Thurston. The Jacobian algebra $A(T):=\mathcal{P}(Q(T), W(T))$
corresponding to the non-degenerate potential $W(T)$ defined by Cerulli Irelli
and the second author is tame, as shown by Schr\"{o}er and the first two
authors. In this paper, we show that there is a natural isomorphism
$\pi_T:\operatorname{Lam}(\boldsymbol{\Sigma})\rightarrow\operatorname{DecIrr}^\tau(A(T))$
of tame partial KRS-monoids that intertwines dual shear coordinates with
respect to $T$, and generic $g$-vectors of irreducible components. Here,
$\operatorname{Lam}(\boldsymbol{\Sigma})$ is the set of laminations of
$\boldsymbol{\Sigma}$ considered by Musiker-Schiffler-Williams, with the
disjoint union of non-intersecting laminations as partial monoid operation. On
the other hand, $\operatorname{DecIrr}^\tau(A(T))$ denotes the set of
generically $\tau$-reduced irreducible components of the decorated
representation varieties of $A(T)$, with the direct sum of generically
$E$-orthogonal irreducible components as partial monoid operation, where $E$ is
the symmetrized $E$-invariant of Derksen-Weyman-Zelevinsky,
$E(-,\bullet)=\dim\operatorname{Hom}_{A(T)}(-,\tau(\bullet))+\dim\operatorname{Hom}_{A(T)}(\bullet,\tau(-))$.
