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There are multiple classes of triangulated categories arising from marked surfaces whose spaces of stability conditions are described by moduli spaces of quadratic differentials on the surfaces. We unify the approaches for describing their spaces of stability conditions and apply this to new classes of examples. This generalizes the results of Bridgeland--Smith to quadratic differentials with arbitrary singularity type (zero/pole/exponential). The novel examples include the derived categories of relative graded Brauer graph algebras. The main computational tool are perverse schobers, which allow us to relate hearts of -structures to mixed-angulations of the surface and tilts of the former with flips of the latter. This is complemented by another approach based on deforming Fukaya -categories of surfaces and transfers of stability conditions.