In this article we show that any hyperbolic Inoue surface (also called Inoue-Hirzebruch surface of even type) admits anti-self-dual bihermitian structures. The same result holds for any of its small deformations as far as its anti-canonical system is non-empty. Similar results are obtained for parabolic Inoue surfaces. Our method also yields a family of anti-self-dual hermitian metrics on any half Inoue surface. We use the twistor method of Donaldson-Friedman  for the proof. -
A., F., Pontecorvo, M. (2010). Anti-self-dual bihermitian structures on Inoue surfaces. JOURNAL OF DIFFERENTIAL GEOMETRY, 85, 15-71.