  Relating different global neutron-star (NS) properties, such as tidal
deformability and radius, or mass and radius, requires an equation of state
(EoS). Determining the NS EoS is therefore not only the science goal of a
variety of observational projects, but it also enters in the analysis process;
for example, to predict a NS radius from a measured tidal deformability via
gravitational waves (GW) during the inspiral of a binary NS merger. To this
aim, it is important to estimate the theoretical uncertainties on the EoS, one
of which is the possible bias coming from an inconsistent treatment of the
low-density region; that is, the use of a so called non-unified NS crust. We
propose a numerical tool allowing the user to consistently match a
nuclear-physics informed crust to an arbitrary high-density EoS describing the
core of the star. We introduce an inversion procedure of the EoS close to
saturation density that allows users to extract nuclear-matter parameters and
extend the EoS to lower densities in a consistent way. For the treatment of
inhomogeneous matter in the crust, a standard approach based on the
compressible liquid-drop (CLD) model approach was used in our work. A Bayesian
analysis using a parametric agnostic EoS representation in the high-density
region is also presented in order to quantify the uncertainties induced by an
inconsistent treatment of the crust. We show that the use of a fixed,
realistic-but-inconsistent model for the crust causes small but avoidable
errors in the estimation of global NS properties and leads to an
underestimation of the uncertainties in the inference of NS properties. Our
results highlight the importance of employing a consistent EoS in inference
schemes. The numerical tool that we developed to reconstruct such a
thermodynamically consistent EoS, CUTER, has been tested and validated for use
by the astrophysical community.
