The quantum dynamics of vibronically interacting systems with additional effects of spin-orbit coupling is studied theoretically. The combined effects of spin-orbit and vibronic coupling lead to geometric phases which can vary smoothly between the case of uncoupled potential energy surfaces (vanishing geometric phase) and the limiting case pertinent for conically intersecting surfaces (geometric phase equals pi). The impact of these phases on the nuclear energy levels and wave functions is revealed by companion studies for two and three coupled potential energy surfaces including and suppressing the geometric phase effects. For free pseudorotation the resulting effective angular momentum can take any noninteger value. The stationary wave functions exhibit a corresponding smooth transition between the limiting cases of integer and half-odd integer angular momentum. For localized wave packets with high angular momentum the geometric phase increases in the same way as found when treating the nuclear coordinates as classical variables. For delocalized wave packets destructive interference leads to a complete loss of symmetry and, for longer times, to a different overall shape of the wave packet. The effects can be relevant also when the nonadiabatic coupling to the higher potential energy surface is negligible.