  Although the spinor field in (1+1) dimensions has the right structure to
model a dispersive bimodal system with gain and loss, the plain addition of
gain to one component of the field and loss to the other one results in an
unstable dispersion relation. In this paper, we advocate a different recipe for
the $\mathcal{PT}$-symmetric extension of spinor models --- the recipe that
does not produce instability of the linear Dirac equation. Having exemplified
the physical origins of the $\mathcal P$- and $\mathcal T$-breaking terms, we
consider the extensions of three U(1)-invariant spinor models with cubic
nonlinearity. Of these, the \PT-symmetric extension of the Thirring model is
shown to be completely integrable and possess infinitely many conserved
quantities. The \PT-symmetric Gross-Neveu equation conserves energy and
momentum but does not conserve charge. The third model is introduced for the
purpose of comparison with the previous two; its \PT-symmetric extension has no
conservation laws at all. Despite this dramatic difference in the integrability
and conservation properties, all three \PT-symmetric models are shown to have
exact soliton solutions. Similar to the solitons of the extended Thirring and
Gross-Neveu equations, the solitons of the new model are found to be stable ---
except for a narrow band of frequencies adjacent to the soliton existence
boundary. The persistence under the $\mathcal P$- and $\mathcal T$-breaking
perturbations as well as the prevalence of stability highlight a remarkable
sturdiness of spinor solitons in (1+1) dimensions.
