Apologies to Edward Said's ghost.
What would the estabilished political regime of a successful Egyptian Expedition after 1798 look like?
Assume that the French naval squadron is not defeated by Nelson at the Battle of the Nile. Because of the continued threat of the French Mediterranean navy, the Ottoman sultan begrudgingly accepts a French presence in Egypt to replace the Mamelukes, and, as a result, Napoleon is not forced to invade Syria, at least for the time being.
Would Napoleon still return to France? Who would be the head of state of this new "Egyptian Republic"? Would Kleber be a good option? If not, then what local authority could they prop up?
What sort of symbology and national ideology would this state adopt? I'd wager that the new French and francophile elite would have to tread careful ground not to offend local Muslim culture. What would be their relationship with the Ottoman sultan, and how would things develop from there?
How would this affects the Jews in the area?
 
The French tried to co-operate with the local elites and introduce Republican elements and Enlightenment ideas into the society. Not surprisingly, nobody was really that keen on going along with it as, unlike in Italy and the Germanies, it had literally no roots anywhere in Egyptian history, society, or intellectual thought. Now, Selim III was somewhat reform minded, and if the French were willing to be clients to HIM rather than the government back home (a possible case, given the coup plotting that's about to hit France proper...) I could see Nappy staying behind and treading an "Islamic Enlightenment" line as a vassal of Constantinople if he's willing to convert (quite likely, given he's not a man of strong faith and great political ambitions) and work in tandum with Ottoman officials to implement and enforce new policies alongside his men (Though THEIR willingness to obey is a little more in question... they might not have much of a choice if they're considered persons non grata by the new French government though).

I could even see him trying to rise up the ranks by offering to use a combined Franco-Egyptian army reformed along European lines to back up the Nizam Cid in allowing the Sultan to crack down on the Janissaries, rather than end up losing the stand off with the Corps during the Edrine Event. Not losing the New Order troops and having another army he could pull up, backed with the "calvary doesn't mutiny" Saphis, he could head off the coup against him with a show of strength and put the Turks on strong footing. Nappy, of course, would be rewarded for his loyalty and assistance with a high rank in the administration...
 
Here’s another idea: what if Napoleon decided to go back to Egypt at any time after his failed expedition? For instance, at the height of his empire.
 
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