  Motivated by recent neutron scattering experiments, we derive and study an
effective "pseudo-dipolar" spin-1/2 model for the XY pyrochlore antiferromagnet
Er2Ti2O7. While a bond-dependent in-plane exchange anisotropy removes any
continuous symmetry, it does lead to a one-parameter `accidental' classical
degeneracy. This degeneracy is lifted by quantum fluctuations in favor of the
non-coplanar spin structure observed experimentally -- a rare experimental
instance of quantum order by disorder. A non-Goldstone low-energy mode is
present in the excitation spectrum in accordance with inelastic neutron
scattering data. Our theory also resolves the puzzle of the experimentally
observed continuous ordering transition, absent from previous models.
