  A scalar field nonminimally coupled to gravity is studied in the canonical
framework, using self-dual variables. The corresponding constraints are first
class and polynomial. To identify the real sector of the theory, reality
conditions are implemented as second class constraints, leading to three real
configurational degrees of freedom per space point. Nevertheless, this
realization makes non-polynomial some of the constraints. The original complex
symplectic structure reduces to the expected real one, by using the appropriate
Dirac brackets. For the sake of preserving the simplicity of the constraints,
an alternative method preventing the use of Dirac brackets, is discussed. It
consists of converting all second class constraints into first class by adding
extra variables. This strategy is implemented for the pure gravity case.
