  We analyse the structure of N=1 and N=2 supersymmetric non-linear
sigma-models built up with a pair of real superfields defined in the superspace
of Atiyah-Ward space-time. The geometry arising has new features such as the
existence of a locally product structure (N=1 case) and a set of automorphisms
of the tangent space that is isomorphic to the split-quaternionic algebra (N=2
case).
