  We prove a criterion for the reality of irreducible representations of the
rational Cherednik algebras H_{0,1}(S_n). This is shown to imply a criterion
for the real loci of the Calogero-Moser spaces C_n in terms of the
Etingof-Ginzburg finite maps \Upsilon \colon C_n \to C^n/S_n \times C^n/S_n,
recovering a result of Mikhin, Tarasov, and Varchenko [MTV2]. As a consequence
we obtain a criterion for the real locus of the Wilson's adelic Grassmannian of
rank one bispectral solutions of the KP hierarchy. Using Wilson's first
parametrisation of the adelic Grassmannian, we give a new proof of a result of
[MTV2] on real bases of spaces of quasi polynomials. The Shapiro Conjecture for
Grassmannians is equivalent to a special case of our result for Calogero-Moser
spaces, namely for the fibres of \Upsilon over C^n/S_n \times 0.
