  Studying the properties of ultra-dense matter is one of the key goals of
modern neutron star research. The measurement of the tidal deformability from
the inspiral of a binary neutron star merger offers one promising method for
constraining the equation of state (EoS) of cold, dense matter. In this work,
we report on a new class of EoSs which have significantly different pressures
at nuclear densities and large differences in stellar radii, but that predict
surprisingly similar tidal deformabilities across the entire range of
astrophysically-observed neutron star masses. Using a survey of 5 million
piecewise polytropic EoSs, subject to five different sets of nuclear priors, we
demonstrate that these "tidal deformability doppelgangers" occur generically.
We find that they can differ substantially in the pressure (by up to a factor
of 3 at nuclear densities) and in the radius of intermediate-mass neutron stars
(by up to 0.5 km), but are observationally indistinguishable in their tidal
deformabilities ($\Delta\Lambda < 30$) with the sensitivity of current
gravitational wave detectors. We demonstrate that this near-degeneracy in the
tidal deformability is a result of allowing for a phase transition at low
densities. We show that a combination of input from nuclear theory (e.g., from
chiral effective field theory), X-ray observations of neutron star radii,
and/or the next generation of gravitational wave detectors will be able to
significantly constrain these tidal deformability doppelgangers.
