  We investigate the geometric phase or Berry phase (BP) acquired by a
spin-half which is both subject to a slowly varying magnetic field and
weakly-coupled to a dissipative environment (either quantum or classical). We
study how this phase is modified by the environment and find that the
modification is of a geometric nature. While the original BP (for an isolated
system) is the flux of a monopole-field through the loop traversed by the
magnetic field, the environment-induced modification of the BP is the flux of a
quadrupole-like field. We find that the environment-induced phase is complex,
and its imaginary part is a geometric contribution to dephasing. Its sign
depends on the direction of the loop. Unlike the BP, this geometric dephasing
is gauge invariant for open paths of the magnetic field.
