  The stack of relative splittings of a special Azumaya algebra plays a key
role in the Non-Abelian Hodge Theory for curves in positive characteristics. In
this paper, we define and study an open substack consisting of the so-called
very good splittings. We show that, when using very good splittings, the
Non-Abelian Hodge isomorphism preserves the semistable loci on the Dolbeault
and the de Rham sides. We also show that the stack of very good splittings
admits a quasi-projective tame moduli space. As a consequence, we show that the
derived pushforwards of the intersection complexes by the Hitchin and the de
Rham-Hitchin morphisms are isomorphic and they have isomorphic perverse
cohomology sheaves.
