  In the paper we consider pseudo bihermitian structures - a pair of complex
structures compatible with a pseudo Riemannian metric. As in the positive
definite case we establish its relations with generalized (pseudo) Kaehler
geometry and holomorphic Poisson structures. We provide a list of compact
complex surfaces which could admit such structure and also examples of
bihermitian structures on some of them. We also consider a naturally defined
null plane distribution on generalized pseudo Kaehler 4-manifold and show that
under a mild restriction it determines an Engel structure.
