  When geometric quantization is applied to a manifold using a real
polarization which is "nice enough", a result of Sniatycki says that the
quantization can be found by counting certain objects, called Bohr-Sommerfeld
leaves. Subsequently, several authors have taken this as motivation for
counting Bohr-Sommerfeld leaves when studying the quantization of manifolds
which are less "nice".
  In this paper, we examine the quantization of compact symplectic manifolds
that can locally be modelled by a toric manifold, using a real polarization
modelled on fibres of the moment map. We compute the results directly, and
obtain a theorem similar to Sniatycki's, which gives the quantization in terms
of counting Bohr-Sommerfeld leaves. However, the count does not include the
Bohr-Sommerfeld leaves which are singular. Thus the quantization obtained is
different from the quantization obtained using a Kaehler polarization.
