  Astrophysical observations of neutron stars probe the structure of dense
nuclear matter and have the potential to reveal phase transitions at high
densities. Most recent analyses are based on parametrized models of the
equation of state with a finite number of parameters and occasionally include
extra parameters intended to capture phase transition phenomenology. However,
such models restrict the types of behavior allowed and may not match the true
equation of state. We introduce a complementary approach that extracts phase
transitions directly from the equation of state without relying on, and thus
being restricted by, an underlying parametrization. We then constrain the
presence of phase transitions in neutron stars with astrophysical data. Current
pulsar mass, tidal deformability, and mass-radius measurements disfavor only
the strongest of possible phase transitions (latent energy per particle
$\gtrsim 100\,\mathrm{MeV}$). Weaker phase transitions are consistent with
observations. We further investigate the prospects for measuring phase
transitions with future gravitational-wave observations and find that catalogs
of \result{$O(100)$} events will (at best) yield Bayes factors of $\sim 10:1$
in favor of phase transitions even when the true equation of state contains
very strong phase transitions. Our results reinforce the idea that neutron star
observations will primarily constrain trends in macroscopic properties rather
than detailed microscopic behavior. Fine-tuned equation of state models will
likely remain unconstrained in the near future.
