  When a multi-qubit state evolves under local unitaries it may obtain a
geometric phase, a feature dependent on the geometry of the state's projective
Hilbert space. A correction term to this geometric phase in addition to the
local subsystem phases may appear from correlations between the subsystems. We
find this correction term can be characterized completely either by the
entanglement or completely by the classical correlations for several classes of
entangled state. States belonging to the former set are W states and their
mixtures, while members of the latter set are cluster states, GHZ states and
two classes of bound entangled state. We probe the structures of these states
more finely using local invariants and suggest the cause of the entanglement
correction is a gauge field like $SL(2,\mathbb{C})$ invariant recently
introduced named twist.
