It turns out that Wikipedia, the final arbiter of knowledge in our current age, has an article on this, calling it the Troupe System. Wikipedia summarizes the Troupe System this way: A Troupe system is a way of playing role-playing games which spreads the game master's responsibilities among each of the players. The term was coined in Ars Magica. It is also known as collaborative role-playing, a term used by other games with a similar mechanism.
Thanks to +Chris C. (The Clash of Spear on Shield) for sharing the Wikipedia link with me, as well as a few other links. Here are some other links:
Troupe System (Wikipedia)
Troupe Style GMing and the Gaming Charter (Gnome Stew)
Troupe Style (Project Redcap)
Starting To Collaborate (Collaborative Roleplay)
And my own blog post from yesterday was a lead-in to this post: Microscope and the Books of Bardo