It is pretty weird and I am having trouble thinking of the smaller things. But so far what I've thought of mrchanically is:

Anomalies will be taken from the SCPs currently on the wiki at my choosing and I'll introduce them to the world a bit at a time so it isn't a total free-for-all at the start. Some SCPs may also need to be modified to either fit with the narrative or to be more manageable such as by shifting dates or adjusting their anomalous properties, thus meaning some anomalies will not be 1:1 copies of their Foundation counterparts.

Each organization will be given rolls or starting parameters and numerical stats to characterize their capabilities in financial resources, manpower, influence, etc. Each may also be given a starting anomaly to play with at the player's behest.

Certain events will increase "heat" and it is in all organizations' interest to maintain the masquerade.

Containing anomalies will consist of the GM putting the news out there that "oh look at this weird thing that does x, y, and z" and interested organizations will have to ID each and then write up a containment narrative that fits with the description, prescribed containment procedures, and the resources available to the group. 

The brainstorming process; however, is certainly still very open and I project it'll remain so for about a month. So what do you think?