As indicated in Eq. ( ), the geometric phase arising in the exact factorization is calculated using the TDVP. While the TDVP is a gauge-dependent quantity, its curl and its circulation are not, thus they are physical observables. In Fig. , we show the curl of the TDVP at three different times along the simulated dynamics, and we compare it to the curl of the nonadiabatic coupling vectors of the Born-Huang representation. Note that in the particular case of a two-state system with real ) and ( adiabatic eigenstates, γ BO (Γ) of Eq. ( ) can be calculated by replacing A BO ν (R) with the nonadiabatic coupling vectors. Therefore, we find instructive to compare the two quantities, i.e., the curl of the TDVP and the curl of the coupling, that give rise to the geometric phases. In Fig. it is clear that the two quantities are qualitatively different, one being smooth everywhere in space and the other presenting a particular behavior only at the location of the conical intersection. Furthermore, we observe that the TDVP is not irrotational and, thus, cannot be eliminated by a suitable choice of gauge. It is clearly dependent on the dynamics of the electron-nuclear system while the nonadiabatic coupling vector is an intrinsic property of the system Hamiltonian itself.