  Following previous work in vacuum spacetimes, we investigate the
constraint-damping properties in the presence of matter of the recently
developed traceless, conformal and covariant Z4 (CCZ4) formulation of the
Einstein equations. First, we evolve an isolated neutron star with an ideal gas
equation of state and subject to a constraint-violating perturbation. We
compare the evolution of the constraints using the CCZ4 and
Baumgarte-Shibata-Shapiro-Nakamura-Oohara-Kojima (BSSNOK) systems. Second, we
study the collapse of an unstable spherical star to a black hole. Finally, we
evolve binary neutron star systems over several orbits until the merger, the
formation of a black hole, and up to the ringdown. We show that the CCZ4
formulation is stable in the presence of matter and that the constraint
violations are one or more orders of magnitude smaller than for the BSSNOK
formulation. Furthermore, by comparing the CCZ4 and the BSSNOK formulations
also for neutron star binaries with large initial constraint violations, we
investigate their influence on the errors on physical quantities. We also give
a new, simple and robust prescription for the damping parameter that removes
the instabilities found when using the fully covariant version of CCZ4 in the
evolution of black holes. Overall, we find that at essentially the same
computational costs the CCZ4 formulation provides solutions that are stable and
with a considerably smaller violation of the Hamiltonian constraint than the
BSSNOK formulation. We also find that the performance of the CCZ4 formulation
is very similar to another conformal and traceless, but noncovariant
formulation of the Z4 system, i.e. the Z4c formulation.
