  The last few years have seen tremendous progress in the observation of the
global properties of neutron stars (NSs), e.g. masses, radii and tidal
deformabilities. Such properties provide information about possible phase
transitions in the inner cores of NSs, provided the connection between observed
masses and radii and the equation of state (EoS) is well understood. We focus
the present study on first-order phase transitions, which often softens the EoS
and consequently reduces the maximum mass as well as the radii of NSs. Here, we
challenge this conventional expectation by constructing explicit examples of
EoSs undergoing a first-order phase transition, but which are much stiffer that
their purely hadronic counterparts. We also provide comparisons with the
recently proposed quarkyonic EoS which suggests a strong repulsion in the core
of NSs, and we show that their stiffness can be realistically masqueraded by
first-order phase transitions to exotic matter.
