This should have no impact on the bode plot but in some of these special cases should make the system come out of saturation more gracefully. I only see this coming into play in a couple instances.
First if the limit changes while the system is saturated it would correct the integrator accordingly.
Second if its a large system with a lag time between the PI controller and the system component that detects the limit. In this case the 'extra' windup can be backed out.
EDIT: Third if the system takes large steps and may jump well over a limit.
Any others?
I design systems with such a lag because I design generic controllers whose output needs to be scaled etc before going to a modulator block which understands and applies limits. There are a handful of clock delays in this path which allow the integrators to wind up a little extra. Though I haven't seen evidence it causes problems.