  Anti-Hermitian mass terms are considered, in addition to Hermitian ones, for
PT-symmetric complex-scalar and fermionic field theories. In both cases, the
Lagrangian can be written in a manifestly symmetric form in terms of the
PT-conjugate variables, allowing for an unambiguous definition of the equations
of motion. After discussing the resulting constraints on the consistency of the
variational procedure, we show that the invariance of a non-Hermitian
Lagrangian under a continuous symmetry transformation does not imply the
existence of a corresponding conserved current. Conserved currents exist, but
these are associated with transformations under which the Lagrangian is not
invariant and which reflect the well-known interpretation of PT-symmetric
theories in terms of systems with gain and loss. A formal understanding of this
unusual feature of non-Hermitian theories requires a careful treatment of
Noether's theorem, and we give specific examples for illustration.
