I see in team instances as well when I do a TI with my alts. For example my alt on their screen might be "ahead" of me and killed a certain ninja whereas I'm a few moves back. And so when I initiate a mystery to target a certain mystyery of what appears before my alt, my alt goes first because what is shown on my screen is not real time.

That siad, the good thiing about this is that it lets me sometimes beat people with much higher initiative which is funny since those people spent so much money on iniatiive.