  We study the problem of optimizing the Shannon mutual information for sources
of real quantum states i.e. sources for which there is a basis in which all the
states have only real components. We consider in detail the sources ${\cal
E}_M$ of $M$ equiprobable qubit states lying symmetrically around the great
circle of real states on the Bloch sphere and give a variety of explicit
optimal strategies. We also consider general real group-covariant sources for
which the group acts irreducibly on the subset of all real states and prove the
existence of a real group-covariant optimal strategy, extending a theorem of
Davies (E. B. Davies, IEEE. Inf. Theory {\bf IT-24}, 596 (1978)). Finally we
propose an optical scheme to implement our optimal strategies, enough simple to
be realized with present technology.
