  A sterile neutrino in the $3+1$ scheme, where the sterile accounts for
neutrino anomalies not explained solely by the weak active neutrinos, arises as
a natural source for the breaking of the $\mu-\tau$ symmetry suggested by
oscillation neutrino data. We explore the predictions for the Dirac CP phases
in this scenario, with and without sterile neutrino decay, and show that
current limits on $\delta_{CP}$ suggest a normal hierarchy and a lightest
neutrino scale below 0.1~eV as the most plausible explanation for that, when
Majorana phases are null. Other Dirac phases turn out to be non zero as well.
