  A real 3-manifold is a smooth 3-manifold together with an orientation
preserving smooth involution, called a real structure. In this article we study
open book decompositions on smooth real 3-manifolds that are compatible with
the real structure. We call them real open book decompositions. We show that
each real open book carries a real contact structure and two real contact
structures supported by the same real open book decomposition are equivariantly
isotopic. We also show that every real contact structure on a closed
3-dimensional real manifold is supported by a real open book. Finally, we
conjecture that two real open books on a real contact manifold supporting the
same real contact structure are related by positive real stabilizations and
equivariant isotopy and that the Giroux correspondence applies to real
manifolds as well namely that there is a one to one correspondence between the
real contact structures on a real 3-manifold up to equivariant contact isotopy
and the real open books up to positive real stabilization. Meanwhile, we study
some examples of real open books and real Heegaard decompositions in lens
spaces.
