i'd say it would need to be a relatively small secret society, namely the leaders and other most influential people of the most powerful countries in the world--let's say G7 at least, G20 at most--who operate in tandem with each other. the best fictional example i can think of is the four major WW2 leaders in the expanded universe of Assassin's Creed (i know i use that series in example alot, but i'd say it's a credible comparison). in that series' lore, FDR, Churchill, Hitler, and Stalin are all Templars, and World War 2 was actually just a front for them to move a Piece of Eden from Point A to Point B, all the way up to Hitler faking his death and leaving a body double in the Fuhrerbunker to be found and destroyed by Soviet soldiers (don't worry, though--the Assassins intercepted him shortly thereafter and killed him anyway)
getting to the point, a realistic global conspiracy would be one that has de jure control over the world if not de facto. for the most part they would be unstoppable, but wouldn't really have absolute control over the entire world--just enough that they would still be the unquestioned masters of the world.