  We propose a general method for deformation quantization of any second-class
constrained system on a symplectic manifold. The constraints determining an
arbitrary constraint surface are in general defined only locally and can be
components of a section of a non-trivial vector bundle over the phase-space
manifold. The covariance of the construction with respect to the change of the
constraint basis is provided by introducing a connection in the ``constraint
bundle'', which becomes a key ingredient of the conversion procedure for the
non-scalar constraints. Unlike in the case of scalar second-class constraints,
no Abelian conversion is possible in general. Within the BRST framework, a
systematic procedure is worked out for converting non-scalar second-class
constraints into non-Abelian first-class ones. The BRST-extended system is
quantized, yielding an explicitly covariant quantization of the original
system. An important feature of second-class systems with non-scalar
constraints is that the appropriately generalized Dirac bracket satisfies the
Jacobi identity only on the constraint surface. At the quantum level, this
results in a weakly associative star-product on the phase space.
