  We give a systematic account of the various pictures of KK-theory for real
C*-algebras, proving natural isomorphisms between the groups that arise from
each picture. As part of this project, we develop the universal properties of
KK-theory, and we use CRT-structures to prove that a natural transformation
from F(A) to G(A) between homotopy equivalent, stable, half-exact functors
defined on real C*-algebras is an isomorphism provided it is an isomorphism on
the smaller class of C*-algebras. Finally, we develop E-theory for real
C*-algebras and use that to obtain new negative results regarding the problem
of approximating almost commuting real matrices by exactly commuting real
matrices.
