  We discuss non-Hermitian field theories where the spectrum of the Hamiltonian
involves only real energies. We make three observations. (i) The theories
obtained from supersymmetric theories by nonanticommutative deformations belong
in many cases to this class. (ii) When the deformation parameter is small, the
deformed theory enjoys the same supersymmetry algebra as the undeformed one.
Half of the supersymmetries are manifest and the existence of another half can
be deduced from the structure of the spectrum. (iii) Generically, the
conventionally defined S--matrix is not unitary for such theories.
