Bihermitian complex surfaces are oriented conformal four-manifolds
admitting two independent compatible complex structures.
Non-anti-self-dual bihermitian structures on ${\mathbb R}^4$
and the four-dimensional torus $T^4$ have recently been
discovered by P. Kobak. We show that an oriented compact 4-manifold,
admitting a non-anti-self-dual bihermitian structure, is a torus
or K3 surface in the strongly bihermitian case (when
the two complex structures are independent at each point) or, otherwise,
must be obtained from the complex projective plane or a minimal ruled
surface of genus less than 2 by blowing up points along some
anti-canonical divisor (but the actual existence of bihermitian structures
in the latter case is still an open question). The paper includes a
general method for constructing non-anti-self-dual bihermitian structures
on tori, K3 surfaces and $S^1\times S^3$. Further properties of compact
bihermitian surfaces are also investigated.
1991 Mathematics Subject Classification: 53C12, 53C55, 32J15.