A drip will highly likely prevent you from having a natural third stage also. Because your uterus is under the pump from the synto, they want to keep it going so you don't bleed. Its truly a cascade as they describe it, one thing leads to so many others.
My third stage of my very fast first labour consisted of my OB asking me to cough, and the placenta came out easily.

I was induced (due to PE and low fetal movement), first contraction at 9.00am, yelling for epi at 10.30am, OB came back at 11.00am (to check if my waters had indeed broken), I rested for an hour, pushed for close to an hour with my OB and MW and DS popped out at 1.02pm.