  Fock and Goncharov introduced a family of cluster algebras associated with
the moduli of SL(k)-local systems on a marked surface with extra decorations at
marked points. We study this family from an algebraic and combinatorial
perspective, emphasizing the structures which arise when the surface has
punctures. When k is 2, these structures are the tagged arcs and tagged
triangulations of Fomin, Shapiro, and Thurston. For higher k, the tagging of
arcs is replaced by a Weyl group action at punctures discovered by Goncharov
and Shen. We pursue a higher analogue of a tagged triangulation in the language
of tensor diagrams, extending work of Fomin and the second author, and we
formulate skein-algebraic tools for calculating in these cluster algebras. We
analyze the finite mutation type examples in detail.
