  It is known that in the WKB approximation of multicomponent systems like
Dirac equation or Born-Oppenheimer approximation, an additional phase appears
apart from the Berry phase. So far, this phase was only examined in special
cases, or under certain restrictive assumptions, namely that the eigenspaces of
the matrix or endomorphism valued symbol of the Hamiltonian form trivial
bundles.
  We give a completely global derivation of this phase which does not depend on
any choice of local trivializing sections. This is achieved using a star
product approach to quantization. Furthermore, we give a systematic and global
approach to a reduction of the problem to a problem defined completely on the
different ``polarizations''. Finally, we discuss to what extent it is actually
possible to reduce the problem to a really scalar one, and make some comments
on obstructions to the existence of global quasiclassical states.
