This is not something that can just be roleplayed if you want it to mean anything. Not that I see it to be much more than doing this:
1) Make a religion, with associated goals, rules and suchlike. These could be various, like not being allowed to annihilate fellow member states, offering the assistance and protection of member states forces in wars against "heretics" and so on. Or even providing more general help in any matter of things.
2) Declare yourself the "pope".
3) Find willing supporters, give them a say in the running of things.
4) Crusade fools and force them to convert then give them a say in the running of things too, or take over the land and install a faction there who does want to support the papacy.
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The only hard bit is making it so people actually want to join, so there needs to be benefits to being a member of this group. However if done right and if the papacy has sufficient sway within powerful member states then it could be very beneficial to join. The hardest thing is getting to that point as until such a group has power and influence there is very little reason to join it.
Really it is just a large semi-alliance with full autonomy within member states likely to help one another but not forced to, whom must follow certain rules regarding how you act amongst and against one another to further the causes of the papacy.
It could be interesting, if done right, not that I plan on joining
