  In this paper we study the property of phase retrievability by redundant
sysems of vectors under perturbations of the frame set. Specifically we show
that if a set $\fc$ of $m$ vectors in the complex Hilbert space of dimension n
allows for vector reconstruction from magnitudes of its coefficients, then
there is a perturbation bound $\rho$ so that any frame set within $\rho$ from
$\fc$ has the same property. In particular this proves the recent construction
in \cite{BH13} is stable under perturbations. By the same token we reduce the
critical cardinality conjectured in \cite{BCMN13a} to proving a stability
result for non phase-retrievable frames.
