Real operator algebras have been studied by Bingren Li and others. A question I'd like to know the answer to in this setting is: Does every von Neumann algebra arise as the complexification of a real von Neumann algebra? Here's a paper of Li on the topic:
http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0936-8.pdf
He also has written a book on this, but I haven't looked at that.
I hope this helps!