  We present a Lorentz-symmetry violating Lagrangian for free fermions, which
is local but not Hermitian, whereas the corresponding Hamiltonian is Hermitian
but not local. A specific feature of the model is that the dispersion relation
is relativistic in both the IR and in the UV, but not in an intermediate
regime, set by a given mass scale. The consistency of the model is shown by the
study of properties expected in analogy with the Dirac Lagrangian.
