  In Ashtekar's Hamiltonian formulation of general relativity, and in loop
quantum gravity, Lorentz covariance is a subtle issue that has been strongly
debated. Maintaining manifest Lorentz covariance seems to require introducing
either complex-valued fields, presenting a significant obstacle to
quantization, or additional (usually second class) constraints whose solution
renders the resulting phase space variables harder to interpret in a spacetime
picture. After reviewing the sources of difficulty, we present a Lorentz
covariant, real formulation in which second class constraints never arise.
Rather than a foliation of spacetime, we use a gauge field y, interpreted as a
field of observers, to break the SO(3,1) symmetry down to a subgroup SO(3)_y.
This symmetry breaking plays a role analogous to that in MacDowell-Mansouri
gravity, which is based on Cartan geometry, leading us to a picture of gravity
as 'Cartan geometrodynamics.' We study both Lorentz gauge transformations and
transformations of the observer field to show that the apparent breaking of
SO(3,1) to SO(3) is not in conflict with Lorentz covariance.
