  The internal composition of neutron stars is still an open issue in
astrophysics. Their innermost regions are impervious to light propagation and
gravitational waves mostly carry global aspects of stars, meaning that only
indirect inferences of their interiors could be obtained. Here we assume a
hypothetical future scenario in which an equation of state softening due to a
phase transition is identified and estimate the observational accuracy to
differentiate a sharp phase transition from a smoother one (associated with a
mixed phase/state due to the unknown value of the surface tension of dense
matter) in a region of a hybrid star by means of some electromagnetic and
gravitational wave observables. We show that different transition constructions
lead to similar sequences of stellar configurations due to their shared
thermodynamic properties. In the most optimistic case - a strong quark-hadron
density jump phase transition - radius observations require fractional
uncertainties smaller than $1\%-2\%$ to differentiate mixed states from sharp
phase transitions. For tidal deformabilities, relative uncertainties should be
smaller than $5\%-10\%$. However, for masses around the onset of stable quark
cores, relative tidal deformability differences associated with strong sharp
phase transitions and mixed states could be much larger (up to around
$20\%-30\%$). All the above suggests that 2.5- and 3rd generation gravitational
wave detectors and near-term electromagnetic missions may be able to start
assessing some particular aspects of phase transitions in neutron stars. In
addition, it points to some limitations on the equation of state recovery using
typical neutron star observables and the impact of systematic uncertainties on
modellings of the equation of state of hybrid stars. Finally, we briefly
discuss other observables that may also be relevant for the probe of mixed
states in stars.
