  Conjugation spaces are topological spaces equipped with an involution such
that their fixed points have the same mod $2$ cohomology (as a graded vector
space, a ring, and even an unstable algebra) but with all degrees divided by
two, generalizing the classical examples of complex projective spaces under
complex conjugation. Spaces which are constructed from unit balls in complex
Euclidean spaces are called spherical and are very well understood. Our aim is
twofold. We construct "exotic" conjugation spaces and study the realization
question: which spaces can be realized as real loci, i.e., fixed points of
conjugation spaces. We identify obstructions and provide examples of spaces and
manifolds which cannot be realized as such.
