Star Wars: The Acolyte official trailer

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Mas Amedda probably could in the event of Palpatine's death, execute Order 66. The likelihood that he would do so, however, is very low. Amedda is a toadying lickspittle no one likes with no personal power base, who is kept around by Palpatine due to his mastery of procedure and bureaucracy. This is, after all, the same man who surrenders the Empire post-Jakku. If Palpatine has an 'in case of my death' contingency, it's far more likely he trusts it to Sly Moore instead.

Fair. Another possibility is that he'd just have some sort of Dead Man's Switch, but I can't see him not having some contingency to set off Order 66 after his death. Palpatine is not the kind of man who accepts defeat with grace.

If Order 66 is not issued, everything depends on what the Clones choose. In the wake of Palpatine's death, Mas Amedda will call a special session of the Senate and a new Chancellor will be chosen. Palpatine's faction will still have the votes, so the new Chancellor will be someone they would support, which is most likely to be Tarkin. If Tarkin becomes Chancellor and the clones remain loyal to the government, they the Jedi are getting booted off Coruscant and sent into exile, and Tarkin rules as Chancellor for Life as a one-party-state style autocrat. On the other hand, if the clones pick the Jedi, then the Jedi coup succeeds, the Senate is purged, and Bail Organa becomes Chancellor a year later. Yoda and the Jedi Council believe the clones will pick them. That strikes me as a very high-risk bet, but they are guided by the Force, so maybe it isn't.

I suspect without the chips the Clones end up with divided loyalties. Some battalions stay loyal to their immediate Jedi commanders, some side with the republic. The Wolfpack probably end up siding with Plo Koon, Ki-Adi Mundi probably still gets iced by Bacara, that sort of thing.

In this scenario the most important divisions are the ones on Coruscant who can be mobilized for or against the coup, the 501st and the Coruscant Guard. The 501st are primarily loyal to Anakin, given the choice they'll follow him wherever he leads. The Coruscant Guard, however, are directly commanded by Palpatine and the Senate, and would absolutely side with his faction. I don't know how that shakes out, the Coruscant Guard seem to be a much bigger division than the 501st but the 501st are the best of the best and the Coruscant Guard very much are not. Could go either way.

I'm not sure how Tarkin would become Chancellor given that he isn't a senator, but rather a member of the military.

Tarkin being appointed to de facto head of state is something I could see, and of Sheev's inner circle I think he's the most likely to end up calling the shots in the event of a power vacuum, but he probably wouldn't take the title Chancellor.