  We present both the gauge theoretic description and the numerical
calculations of the Berry phases with the real eigenstates, involving one with
a many-body system as a background and the other with no such background. We
demonstrate that for the former the sign of the Berry phase factor for a spin
$\f{1}{2}$ particle (hole) coupled to a slow subsystem (phonon) depends on both
the strength of electron correlations and the characteristics of the closed
paths, unlike the cases for the latter.
