  We consider a two-level system coupled to a highly non-Markovian environment
when the coupling axis rotates with time. The environment may be quantum (for
example a bosonic bath or a spin bath) or classical (such as classical noise).
We show that an Anderson orthogonality catastrophe suppresses transitions, so
that the system's instantaneous eigenstates (parallel and anti-parallel to the
coupling axis) can adiabatically follow the rotation. These states thereby
acquire Berry phases; geometric phases given by the area enclosed by the
coupling axis. Unlike in earlier proposals for environment-induced Berry
phases, here there is little decoherence, so one does not need a
decoherence-free subspace. Indeed we show that this Berry phase should be much
easier to observe than a conventional one, because it is not masked by either
the dynamic phase or the leading non-adiabatic phase. The effects that we
discuss should be observable in any qubit device where one can drive three
parameters in the Hamiltonian with strong man-made noise.
