Well, as Moose already mentioned, Ul'dah's government is kind of complicated - officially and legally it's a Sultanate with an autocratic head of state (as in, Nanamo), but, the Sultana's actual power had been eroded years ago due to the merchants and businessmen and women who keep the gil flowing into the city (thus can easily cut said trade off and paralyze the city) becoming the real power behind the throne, thus the nation is more accurately and effectively an oligarchy/plutocracy. The Sultana still has some sway, but she seems to be little more than a compliant rubber stamp on legislation and thus cannot resolve deadlocks (This was shown in 2.2 when the Syndicate voted not to help the Domans and Nanamo could not interfere).
This coupled with the fact the only remaining Royalist on the Syndicate is Raubahn (although Lolorito despite being a Monetarist still does secretly support the Sultana, at least trying to keep the status quo in place - I mean, he could have seized control of the city following the whole banquet affair but instead he conspired to create a complicated Batman gambit to eliminate Teledji and protect the Sultana and her position, and Dewlala is more neutral), it's not hard to see how the Syndicate managed to rival or even exceed the power of the traditional Sultanate.
Even after that though, Raubahn is also effectively the military commander of Ul'dah, as his position of leader of the Immortal Flames, Ul'dah's Grand Company. Unlike the other city states, until the Grand Companies were reformed, Ul'dah actually did not have a permanent standing professional army - there was the elite unite of knights (that is, paladins) the Sultansworn who served as the personal royal guard of the Sultana, but apart from that there was no actual 'Ul'dah Army' or anything, instead the Sultanate made use of non-professional mercenaries and sellswords grouped under several individual units (the Brass Blades and the Stone Torches are examples of this). Because such mercenaries were directly paid by members of the Syndicate for their service instead of the Sultanate, it's not hard to see how their loyalties towards the throne might turn out to be rather questionable.
Accordingly, when the Immortal Flames were established just prior to the Calamity, this led to further tension in the Syndicate as the Royalists now had an effective army of their own and could theoretically rival the individual private armies of the Syndicate if things ever took a turn for the worse (which very nearly happened after the 'assassination', but thankfully it didn't come to that).
Either way, in regards to Raubahn and Nanamo's relationship and who is the actual leader of Ul'dah, think of it like this - Nanamo is the legal ruler, the Sultana, and Raubahn is her military leader, the one responsible for defending the city and representing the city as head of their Grand Company in the Alliance, and that together the two share power of ruling the citystate. Please feel free to clarify or criticize any mistakes I've inevitably made though!
Maybe that is indeed the case - remember back to when the Warrior of Light is fleeing Ul'dah after the banquet and first encounters Pipin - he states that he's been "at the Ala Mhigian Front" for an extended period, strongly suggesting that the Immortal Flames at least are undertaking some kind of long-term military action at Ala Mhigo against the Garlean occupation - given Raubahn's Ala Mhigan origins this would just would make perfect sense for something that Nanamo herself would support (as the Immortal Flames are rather stretched personnel and resources wise, the fact he was able to dedicate manpower to such an operation seems to make that even more likely). But nothing is known for certain other than this statement by Pipin so take from that what you will.![]()