  Let X be an Enriques surface over the field of complex numbers. We prove that
there exists a nontrivial quaternion algebra A on X. Then we study the moduli
scheme of torsion free A-modules of rank one. Finally we prove that this moduli
scheme is an \'{e}tale double cover of a Lagrangian subscheme in the
corresponding moduli scheme on the associated covering K3 surface.
