The geometric phases, γ BO (Γ) and γ(Γ,t), have been calculated from Eqs. ( ) and ( ), respectively, and are reported in Table . We used different paths Γ, namely Γ CI1 and Γ CI2 encircling the position of the conical intersection, and Γ X and Γ Y that do not encircle the position of the conical intersection. The paths are shown in Fig. . Table shows that in the adiabatic representation, namely for for γ BO (Γ), a value close to π is obtained only for Γ CI1 and Γ CI2 , and that we find a value much smaller, i.e., close to zero, along the other paths. Using the TDVP, all the values change with time and with the path. Thus, the dynamics-induced geometric phase arising in the exact factorization is not a topological feature but rather caused and governed by the dynamics of the system, in stark contrast with the topological, path-independent phase that arises only in the adiabatic representation. 0.24 ms 0.69 ms 1.35 ms adiabatic Γ CI1 -0.0209 0.0265 0.0506 3.12 Γ CI2 -0.794 2.05 -0.00973 3.12 Γ X 0.00269 0.0624 0.000178 -0.00281 Γ Y -0.758 0.157 0.188 0.000581 TABLE I. Geometric phase γ(Γ,t) at three times throughout the dynamics (first three columns) along four different paths, encircling the point representing the location of the conical intersection (Γ CI1 , Γ CI2 ) and without encircling this point (Γ X , Γ Y ). The last column reports the values of γ BO (Γ) using the same paths.