Every complex manifold carries a canonical orientation, and it is natural to wonder when the underlying topological or smooth manifold carries a complex structure compatible with the other orientation. In [2], Beauville raised this question for compact complex surfaces. He noted that there are a lot of examples, like products of curves, or Hopf surfaces, where the underlying manifold admits an orientation-reversing selfdiffeomorphism. This implies that the signature is zero. Beauville asked if there are any examples of non-zero signature.