  We extend the notion of ascent-compatibility from symmetric groups to all
Coxeter groups, thereby providing a type-independent framework for constructing
families of modules of $0$-Hecke algebras. We apply this framework in type $B$
to give representation-theoretic interpretations of a number of noteworthy
families of type-$B$ quasisymmetric functions. Next, we construct modules of
the type-$B$ $0$-Hecke algebra corresponding to type-$B$ analogues of Schur
functions and introduce a type-$B$ analogue of Schur $Q$-functions; we prove
that these shifted domino functions expand positively in the type-$B$ peak
functions. We define a type-$B$ analogue of the $0$-Hecke--Clifford algebra,
and we use this to provide representation-theoretic interpretations for both
the type-$B$ peak functions and the shifted domino functions. We consider the
modules of this algebra induced from type-$B$ $0$-Hecke modules constructed via
ascent-compatibility and prove a general formula, in terms of type-$B$ peak
functions, for the type-$B$ quasisymmetric characteristics of the restrictions
of these modules.
