To understand the underlying differences between the OP and MP protocols, let us focus on the trivial crossing at t = 110.5 fs as labeled by the black square (■) in Fig. . We observe that, for an incoming superposition state wavefunction, the electronic transition between states 74 (orange line) and 75 (magenta line) affects the relative phases of the other states (here 84 blue line). For dt c = 0.24 fs at (■), the overlap matrix following the OP protocol is (for states 74, 75 and 84) Here we notice that the MP protocol flips the sign of column 74 of the overlap matrix to ensure det{U M P } = 1, leading to changing the sign relative to states 84. (The two protocols also differ by a sign in column 122 at this point in time, but this adiabat is not populated and so is of no consequence.) While this relative sign discrepancy does not lead to a difference in populations for 10 fs (i.e. until t = 120 fs), this difference will eventually lead to dramatic differences when yet another crossing is encountered.