  We analyze observations of eight quiescent low-mass X-ray binaries in
globular clusters and combine them to determine the neutron star mass-radius
curve and the equation of state of dense matter. We determine the effect that
several uncertainties may have on our results, including uncertainties in the
distance, the atmosphere composition, the neutron star maximum mass, the
neutron star mass distribution, the possible presence of a hotspot on the
neutron star surface, and the prior choice for the equation of state of dense
matter. We find that the radius of a 1.4 solar mass neutron star is most likely
from 10 to 14 km and that tighter constraints are only possible with stronger
assumptions about the nature of the neutron stars, the systematics of the
observations, or the nature of dense matter. Strong phase transitions are
preferred over other models and interpretations of the data with a Bayes factor
of 8 or more, and in this case, the radius is likely smaller than 12 km.
However, radii larger than 12 km are preferred if the neutron stars have uneven
temperature distributions.
