  Although the WKB approximation for multicomponent systems has been
intensively studied in the literature, its geometric and global aspects are
much less well understood than in the scalar case. In this paper we give a
completely geometric derivation of the transport equation, without using local
sections and without assuming complete diagonalizability of the matrix valued
principal symbol, or triviality of its eigenbundles. The term (called ``no-name
term'' in some previous literature) appearing in the transport equation in
addition to the covariant derivative with respect to a natural projected
connection will be a tensor, independent of the choice of any sections. We give
a geometric interpretation of this tensor, involving the contraction of the
curvature of the eigenbundle and an analog of the second fundamental form with
the Poisson tensor in phase space. In the non-degenerate case this term may be
rewritten in an even simpler geometric form. Finally, we discuss obstructions
to the existence of WKB states and give a geometric description of the
quantization condition for WKB states for a non-degenerate eigenvalue-function.
