  We consider the real dynamics of a two parameter family of plane birational
maps, focusing especially on an open subset of parameter space on which the
real and complex dynamics are in close agreement. On the complex side, we find
a rational complex surface to which the maps extend in a well-defined fashion
and then calculate the induced action on the Picard group of the surface. On
the real side, we use the critical sets of the maps to produce a combinatorial
model for the dynamics. By comparing the two points of view, we are able to
code points in the real non-wandering set, describe the behavior of wandering
orbits, and identify a measure of maximal entropy for the real dynamics.
