  Let h be a Real bundle, in the sense of Atiyah, over a space X. This is a
complex vector bundle together with an involution which is compatible with
complex conjugation. We use the fact that BU is equipped with a structure of
conjugation space, as defined by Hausmann, Holm, and Puppe, to construct
equivariant Chern classes in the Z/2-equivariant cohomology of X with twisted
integer coefficients. We show that these classes determine the
(non-equivariant) Chern classes of h, forgetting the involution on X, and the
Stiefel-Whitney classes of the real bundle of fixed points.
