The observable nature of topological phases related to conical intersections in molecules is studied. Topological phases should be ubiquitous in molecular processes, but their elusive character has often made them a topic of discussion. To shed some light on this issue, we simulate the dynamics governed by a Jahn-Teller Hamiltonian and analyze it employing two theoretical representations of the molecular wave function: the adiabatic and the exact factorization. We find fundamental differences between effects related to topological phases arising exclusively in the adiabatic representation, and thus not related to any physical observable, and geometric phases within the exact factorization that can be connected to an observable quantity. We stress that while the topological phase of the adiabatic representation is an intrinsic property of the Hamiltonian, the geometric phase of the exact factorization depends on the dynamics that the system undergoes and is connected to the circulation of the nuclear momentum field.