  We compare real and complex dynamics for automorphisms of rational surfaces
that are obtained by lifting \chg{some} quadratic birational maps of the plane.
In particular, we show how to exploit the existence of an invariant cubic curve
to understand how the real part of an automorphism acts on homology. We apply
this understanding to give examples where the entropy of the full (complex)
automorphism is the same as its real restriction. Conversely and by different
methods, we exhibit different examples where the entropy is strictly decreased
by restricting to the real part of the surface. Finally, we give an example of
a rational surface automorphism with positive entropy whose periodic cycles are
all real.
