  The paper aims to spell out the relevance of the Berry phase in view of the
question what the minimal mathematical structure is that accounts for all
observable quantum phenomena. The question is both of conceptual and of
ontological interest. While common wisdom tells us that the quantum structure
is represented by the structure of the projective Hilbert space, the
appropriate structure rich enough to account for the Berry phase is the U(1)
bundle over that projective space. The Berry phase is ultimately rooted in the
curvature of this quantum bundle, it cannot be traced back to the Hamiltonian
dynamics alone. This motivates the ontological claim in the final part of the
paper that, if one strives for a realistic understanding of quantum theory
including the Berry phase, one should adopt a form of ontic structural realism.
