  Affine cluster varieties are covered up to codimension 2 by open algebraic
tori. We put forth a general conjecture (based on earlier conversation between
Vivek Shende and the last author) characterizing their deep locus, i.e. the
complement of all cluster charts, as the locus of points with non-trivial
stabilizer under the action of cluster automorphisms. We use the diagrammatics
of Demazure weaves to verify the conjecture for skew-symmetric cluster
varieties of finite cluster type with arbitrary choice of frozens and for the
top open positroid strata of Grassmannians $\mathrm{Gr}(2,n)$ and
$\mathrm{Gr}(3,n)$. We illustrate how this already has applications in
symplectic topology and mirror symmetry, by proving that the Fukaya category of
Grassmannians $\mathrm{Gr}(2,2n+1)$ is split-generated by finitely many
Lagrangian tori, and homological mirror symmetry holds with a Landau--Ginzburg
model proposed by Rietsch. Finally, we study the geometry of the deep locus,
and find that it can be singular and have several irreducible components of
different dimensions, but they all are again cluster varieties in our examples
in really full rank cases.
