<p>We consider configurations consisting of a gravitating nonlinear spinor field and a massless ghost scalar field providing a nontrivial spacetime topology. For such a mixed system, we have constructed families of asymptotically flat asymmetric solutions describing Dirac stars with wormhole topology. The physical properties of such systems are completely determined by the values of three input quantities: the nonlinearity parameter of the spinor field, the spinor frequency, and the throat parameter. Depending on the specific values of these parameters, the configurations may be regular or singular and possess one or two throats, as well as they may contain possible horizons of different kinds. Furthermore, because of the asymmetry, masses and sizes of the configurations observed at the two asymptotic ends of the wormhole may differ considerably. For large negative values of the nonlinearity parameter, it is possible to obtain configurations whose masses are comparable to the Chandrasekhar mass and effective radii are of the order of kilometers.</p>