When the last province is acquired can the person finish their turn or is the game immediately over? It seems like the rules dictate that a player may always finish their turn.
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You can always finish your turn. Whether you picked up that last Province in the action phase or the buy phase, you'll be able to keep playing until you're done with your turn, possibly picking up another victory card or two. Note that the only cards you can possibly find useful at this point are cards that give you victory points. (If you have Gardens, then you might buy any old cards just to make your deck bigger. Similarly, you might pick up Silvers if you have a Feodum.)
As the others have said, you always finish your turn (and thus might get several more victory cards). Generally this comes up if you gain a province with one of your actions, as you could otherwise reorder your buys to get the province last, but it also matters if you're hitting another game end condition as well (say, you buy the last estate, thus emptying the third pile).
One of the equal turns variations is ghost provinces, in which running out of provinces triggers the game end but you add extra provinces so everyone who hasn't had their last turn yet (and the person triggering game end) still gets to buy them.