  We carry out a thorough study of weight-shifting operators on Hilbert modular
forms in characteristic $p$, generalizing the author's prior work with Sasaki
to the case where $p$ is ramified in the totally real field $F$. In particular
we use the partial Hasse invariants and Kodaira-Spencer filtrations defined by
Reduzzi and Xiao to improve on Andreatta and Goren's construction of partial
$\Theta$-operators, obtaining ones whose effect on weights is optimal from the
point of view of geometric Serre weight conjectures. Furthermore we describe
the kernels of partial $\Theta$-operators in terms of images of geometrically
constructed partial Frobenius operators. Finally we apply our results to prove
a partial positivity result for minimal weights of mod $p$ Hilbert modular
forms.
