The Geometry of N=1 and N=2 Real Supersymmetric Non-Linear $\sigma$-Models in the Atiyah-Ward Space-Time

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)