  We develop a unified approach for identifying spaces of stability conditions
of triangulated categories arising from weighted marked surfaces with moduli
spaces of quadratic differentials. This approach is based on the notion of a
perverse schober (perverse sheaf of triangulated categories) and their
triangulated categories of global sections. Under suitable conditions on the
perverse schober, we identify mixed-angulations and their flips with
finite-length hearts and their tilts, which then leads to the identification of
moduli spaces. As an application we obtain a generalization of the results of
Bridgeland--Smith to quadratic differentials with arbitrary singularity type
(zero/pole/exponential).
