  We develop the categorical context for defining Hermitian non-semisimple
TQFTs. We prove that relative Hermitian modular categories give rise to
modified Hermitian WRT-TQFTs and provide numerous examples of these structures
coming from the representation theory of quantum groups and quantum
superalgebras. The Hermitian theory developed here for the modified Turaev-Viro
TQFT is applied to define new pseudo-Hermitian topological phases that can be
considered as non-semisimple analogs of Levin-Wen models.
