  We study how generic phase transitions taking place in compact stars
constructed in the framework of the Eddington-inspired Born-Infeld (EiBI)
gravity can lead to anomalous behavior of these stars. For the case with
first-order phase transitions, compact stars in EiBI gravity with a positive
coupling parameter $\kappa $ exhibit a finite region with constant pressure,
which is absent in general relativity. However, for the case with a negative
$\kappa $, an equilibrium stellar configuration cannot be constructed. Hence,
EiBI gravity seems to impose stricter constraints on the microphysics of
stellar matter. Besides, in the presence of spatial discontinuities in the
sound speed $c_s$ due to phase transitions, the Ricci scalar is spatially
discontinuous and contains $\delta$-function singularities proportional to the
jump in $c_s^2$ acquired in the associated phase transition.
