  In Loop Quantum Gravity, tremendous progress has been made using the
Ashtekar-Barbero variables. These variables, defined in a gauge-fixing of the
theory, correspond to a parametrization of the solutions of the so-called
simplicity constraints. Their geometrical interpretation is however
unsatisfactory as they do not constitute a space-time connection. It would be
possible to resolve this point by using a full Lorentz connection or,
equivalently, by using the self-dual Ashtekar variables. This leads however to
simplicity constraints or reality conditions which are notoriously difficult to
implement in the quantum theory.
  We explore in this paper the possibility of imposing such constraints at the
quantum level in the context of canonical quantization. To do so, we define a
simpler model, in 3d, with similar constraints by extending the phase space to
include an independent vielbein. We define the classical model and show that a
precise quantum theory by gauge-unfixing can be defined out of it, completely
equivalent to the standard 3d euclidean quantum gravity.
  We discuss possible future explorations around this model as it could help as
a stepping stone to define full-fledged covariant Loop Quantum Gravity.
