We study the coupling of $N=1$ supergravity to a class of supersymmetric nonlinear $\ensuremath{\sigma}$ models in an $N=1$ superspace which is based on the Atiyah-Ward space-time of a (2+2)-signature metric. We also discuss the conditions for the implementation of the gauging of isometries of the associated K\"ahlerian target spaces and present the resulting gauge-covariant supersymmetric action functional in the curved superspace. In an appendix, we aim at clarifying some aspects of the ordinary (1+3)-signature version of the very same problem.