  This article is a continuation of a paper of the first author \cite{F} about
complex structures on real Banach spaces. We define a notion of even infinite
dimensional real Banach space, and prove that there exist even spaces,
including HI or unconditional examples from \cite{F} and $C(K)$ examples due to
Plebanek \cite{P}. We extend results of \cite{F} relating the set of complex
structures up to isomorphism on a real space to a group associated to
inessential operators on that space, and give characterizations of even spaces
in terms of this group. We also generalize results of \cite{F} about totally
incomparable complex structures to essentially incomparable complex structures,
while showing that the complex version of a space defined by S. Argyros and A.
Manoussakis \cite{AM} provide examples of essentially incomparable complex
structures which are not totally incomparable.
