  In previous work of the second- and third-named authors with Linhui Shen,
cluster theory was used to construct wavefunctions for branes in threespace and
conjecturally relate them to open Gromov-Witten invariants. This was done by
defining a quantum Lagrangian subvariety of a quantum cluster variety, and
mutating a simple solution to the defining equations in a distinguished seed.
In this paper, we extend the construction to incorporate the skein-theoretic
approach to open Gromov-Witten theory of Ekholm-Shende. In particular, we
define a skein-theoretic version of cluster theory, including the groupoid of
seeds and mutations and a skein-theoretic version of the quantum dilogarithm.
We prove a pentagon relation in the skein of the closed torus in this context,
and give strong evidence that its analogue holds for arbitrary surfaces. We
propose face relations satisfied by the skein-theoretic wavefunction, prove
their invariance under mutations, and show their solution is unique. We define
a skein version of framings in the story, and use the novel cluster structure
to compute wavefunctions in several examples. The skein approach incorporates
moduli spaces of sheaves of higher microlocal rank and their quantizations.
