Religion with a new banner made corporations in Civ4 kind of boring and spammy, increasing the already busy late game.
If corporations make a return, as TheKingofBigOz noted, it should be more integrated in the game mechanics.
For example, it should start in the renaissance with you (the leader) granting charters for groups of people to form a corporation and the type and scope of that corporation will determine how your empire evolves over time.
So in short, my vision of it is that I see it more as a social policy type layer, with potentially units to move around but not the kind of missionary/preachy religion type units, but economic units you can place in another Civ's cities/capitals as a sign of Foreign direct investment or you may choose to invest locally.
At the same time, corporations can exert soft power (tourism) influence spreading your culture.
This would mean a revamp of trade routes/tourism as a whole if they were ever to add it into the post BNW build of the game, assuming there's a 3rd expansion lurking out there.
If added into Civ6 or a future game, Corporations should also be given some level of independence (gradiated from free market to state controlled) such that they will try to influence your foreign and domestic policies under free market scenarios, where the trade off is high gpt income/and other bonus for less control, over state controlled here you have more control but less gpt income.