  Let $F$ be a totally real field of degree $g$, and let $p$ be a prime number.
We construct $g$ partial Hasse invariants on the characteristic $p$ fiber of
the Pappas-Rapoport splitting model of the Hilbert modular variety for $F$ with
level prime to $p$, extending the usual partial Hasse invariants defined over
the Rapoport locus. In particular, when $p$ ramifies in $F$, we solve the
problem of lack of partial Hasse invariants. Using the stratification induced
by these generalized partial Hasse invariants on the splitting model, we prove
in complete generality the existence of Galois pseudo-representations attached
to Hecke eigenclasses of paritious weight occurring in the coherent cohomology
of Hilbert modular varieties $\mathrm{mod}$ $p^m$, extending a previous result
of M. Emerton and the authors which required $p$ to be unramified in $F$.
