  We show that any hyperbolic Inoue surface (or Inoue-Hirzebruch surface of
even type) admits anti-self-dual bihermitian structures. The same result also
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 anti-self-dual hermitian metrics on any half Inoue surface.
We use the twistor method of Donaldson-Friedman for the proof.
