  We verify that a large portion of the theory of complex operator spaces and
operator algebras (as represented by the 2004 book by the author and Le Merdy
for specificity) transfers to the real case. We point out some of the results
that do not work in the real case. We also discuss how the theory and standard
constructions interact with the complexification. For example, we develop the
real case of the theory of operator space multipliers and the operator space
centralizer algebra, and discuss how these topics connect with the
complexification. This turns out to differ in some important details from the
complex case. We also characterize real structure in complex operator spaces;
and give `real' characterizations of some of the most important objects in the
subject.
