  We initiate the study of decorated character stacks and their quantizations
using the framework of stratified factorization homology. We thereby extend the
construction by Fock and Goncharov of (quantum) decorated character varieties
to encompass also the stacky points, in a way that is both compatible with
cutting and gluing and equivariant with respect to canonical actions of the
modular group of the surface. In the cases $G=SL_2,PGL_2$ we construct a system
of categorical charts and flips on the quantum decorated character stacks which
generalize the well--known cluster structures on the Fock--Goncharov moduli
spaces.
