  In this paper we construct an abelian category of "mixed perverse sheaves"
attached to any realization of a Coxeter group, in terms of the associated
Elias-Williamson diagrammatic category. This construction extends previous work
of the first two authors, where we worked with parity complexes instead of
diagrams, and we extend most of the properties known in this case to the
general setting. As an application we prove that the split Grothendieck group
of the Elias-Williamson diagrammatic category is isomorphic to the
corresponding Hecke algebra, for any choice of realization.
