  Using $\rm Er_2Ti_2O_7$ as a motivation, we investigate finite-field
properties of $XY$ pyrochlore antiferromagnets. In addition to a
fluctuation-induced six-fold anisotropy present in zero field, an external
magnetic field induces a combination of two-, three-, and six-fold clock terms
as a function of its orientation providing for a rich and controllable
magnetothermodynamics. For $\rm Er_2Ti_2O_7$, we predict a new phase transition
for ${\bf H}\parallel [001]$. Re-entrant transitions are also found for ${\bf
H}\parallel [111]$. We extend these results to the whole family the $XY$
pyrochlore antiferromagnets and show that presence and number of low-field
transitions for different orientations can be used for locating a given
material in the parameter space of anisotropic pyrochlores. Finite-temperature
classical Monte Carlo simulations serve to confirm and illustrate these
analytic predictions.
