If the field of alternate timeline sociology teaches us anything, it teaches that there are three paths to big-c Conspiracies. There is the accretion path, in which one shadowy group takes over more and more other groups until it discovers, one day, that they’re the Secret Masters now, hooray!; there is the reorganizational path, where a shadowy group attempting accretion discovers that their Conspiracy has been taken over from the inside and/or parceled off into a number of half-independent factions inside the main structure, which then contest amongst themselves for power; and there is the amalgamation path, where various independent shadowy groups deliberately combine efforts to create a Conspiracy built up of sovereign, autonomous entities. That last sort is called a ‘Conglominatti’ by the more vulgar, and is discussed below.
The advantages of a Conglominatti are undeniable. Where an accretion-style Illuminati is vulnerable to becoming a totalitarian state (either bureaucratic or tyrannous), and a reorganizational Illuminati is notoriously prone to becoming a feudal oligarchy (also with elements of totalitarianism), a Conglominatti can balance day-to-day affairs using a functional aristocratic structure that concentrates on its own areas of expertise while allowing other parts of the Conglominatti to get on with their own business. Also, since the strategies and goals of individual groups inside a successful Conglominatti are by definition compatible with each other while not necessarily being identical, it is easier for dissidents inside the Conglominatti to remain productive and unmolested. In fact, given sufficient numbers said dissidents can be elevated into an actual group inside the larger power structure, giving them a buy-in — and an incentive to work with the larger Conspiracy, and not against it. Lastly: the sovereign status of each group can double as a firewall if one particular member group is corrupted or suborned by mundane or esoteric adversaries. More than one Congolminatti has been saved by its inability to be decapitated in a single surgical strike.
Disadvantages? Inefficiency, for a start. Accretion-style Illuminati work more quickly; reorganizational ones weed out stupid Secret Masters in a deliberately Darwinian process. There’s also the problem that Conglominatti are more prone than the other kinds of Conspiracies to having ‘rogue’ groups inside of it. While memetic drift is a problem for any Conspiracy, it can ferment for far longer in a system where a certain amount of divergent thinking is expected on the institutional level. But the largest challenge a Conglominatti can face is a relative lack of ideological unity. Accretionists have the vision of the Secret Masters to ‘guide’ them. Reorganizationalists can usually at least agree on a basic set of working principles. But Conglominatti are the most likely to have a Grand Council or the like, and if the Grand Council falls apart over a dispute then the resulting Shadow War can be particularly nasty; after all, everybody knows how everybody else thinks and operates.
So, which is the best sort for a new Conspiracy? Nobody knows. Or rather everybody knows, only nobody agrees with each other about the answer. This is another thing that alternate timeline sociology teaches us; having more than one universe to draw data from sometimes merely gives the researcher far too many answers to a question.