  Using the idea of a generalized Kaehler structure, which is a pair of
commuting generalized complex structures, we construct bihermitian metrics on
the projective plane and the product of two projective lines, and show that any
such structure on a compact 4-manifold M defines one on the moduli space of
anti-self-dual connections on a fixed principal bundle over M. We highlight the
role of holomorphic Poisson structures in all these constructions.
