  A real 3- or 4-manifold has by definition an orientation preserving smooth
involution acting on it. We consider Lefschetz fibrations of 4-dimensional
manifolds-with-boundary and open book decompositions on their boundary in the
existence of a real structure. We prove that there is a real open book which
cannot be filled by a real Lefschetz fibration, although it is filled by
non-real Lefschetz fibrations.
