So I think I might have cracked this one, again with a lot of help from T.W.Wombat's Historia Obscura, as well as my own Amontillado.

• Players will roll to discover things about themselves.
• Players will roll to discover things about a petitioner, praying for a miracle, and the prayer that they are trying to have answered.
• Players will roll to discover the kind of miracle the petitioner received. It will frequently seem to be out of step with the petition.
• Players will begin to tell a story, one by one, about the petitioner, building hopefully to an explanation for why, in fact, despite all appearances, their prayer was in fact miraculously answered.
• The New Pope will be able to ask questions and cast doubt on the story, rolling dice against the players to decide whether he's convinced. I think, like Amontillado, the New Pope might be rolling 2xD6 against the players' 1.
• A player success will lead to the questioned player to continue to add to the story. A player failure will lead to the players holding a conclave to decide who adds to the story next.  
• Story can't end unless everyone speaks.
• Players can collect tokens to lend additional Credibility to their story against the New Pope's skeptical rolls.
• The thing I'm trying to decide now is how to, ultimately, decide whether the New Pope believes the story or not. Hmm.

Anyway, that's where I am with it.