  Pseudo-Hermitian field theories possess a global continuous ``similarity''
symmetry, interconnecting the theories with the same physical particle content
and an identical mass spectrum. In their regimes with real spectra, within this
family of similarity transformations, there is a map from the non-Hermitian
theory to its Hermitian similarity partner. We promote the similarity
transformation to a local symmetry, which requires the introduction of a new
vector similarity field as a connection in the similarity space of
non-Hermitian theories. In the case of non-Hermitian two-flavour scalar or
fermion mixing, and by virtue of a novel IR/UV mixing effect, the effect of
inhomogeneous non-Hermiticity then reveals itself via anomalous dispersion,
instabilities and superluminal group velocities at very high momenta, thus
setting an upper bound on the particle momentum propagating through
inhomogeneous backgrounds characterised by Lagrangians with non-Hermitian mass
matrices. Such a non-Hermitian extension of the Standard Model of particle
physics, encoded in a weak inhomogeneity of the non-Hermitian part of the
fermion mass matrix, may nevertheless provide us with a low-energy particle
spectrum consistent with experimentally observed properties.
