Generalized complex and paracomplex structures on product manifolds

On a product manifold (M, r), we consider four geometric structures compatible withr, e.g. hyper-paracomplex or bi-Lagrangian, and define distinguished generalized complex or paracomplex structures onM, which interpolate between some pairs of them. We study the twistor bundles whose smooth sections are these new structures, obtaining the typical fibers as homogeneous spaces of classical groups. Also, we give examples of product manifolds admitting some of these new structures.