  We develop a theory of tdos and twisted $\mathcal D$-modules over general
base schemes with a focus on functorial aspects. In particular, we introduce a
flat base change functor and establish its compatibility with globalization and
direct image functors. We also study forms of closed $K$-orbits of
$\theta$-stable parabolic subgroups in the total flag variety. We apply these
two developed theories to give a geometric construction of half-integral models
of cohomologically induced modules. With a view towards arithmetic
applications, we further demonstrate desirable properties of the constructed
half-integral models, such as projectivity over the base and torsion-free
relative Lie algebra cohomology.
