The Taiidan Republic
It is impossible to talk about the Taiidan Republic without first talking about the Taiidan empire. It was a short lived polity, but it was instrumental in shaping much of current galactic society.
During the Bentusi peace the systems that would eventually form the Taiidan empire were controlled by three separate polities. These were the Sur'Kesh Union, Thessian Council, and Tuchankan Tribes. And it was these three more than any other that fomented the end of Bentusi power and rise of the galactic council. The long lives of the Asari matriarchs and Krogan warlords allowed them to amass impressive power bases and with the addition of Salarian intelligence they were able to create a web of conspiracies that spanned the galaxy. With a single message they fomented wars in a hundred systems.
The actual period of chaos and war lasted for approximately two decades before the architects of the plan were able to draw their webs shut and install the galactic council with themselves at the head. Nominally every nation in the galaxy had an equal say, but in practice the three co conspirators coopted most of the council's decisions by unifying as the single largest and most powerful of all the surviving polities, the Taiidan Empire.
It was not long however before the Krogans began to chafe at their place in their order of things. They desired more. More power, more room, more resources, and they felt that if they could not acquire such things peacefully, then they could only take them by force.
Splitting off from the Taiidan empire they formed the Hiigaran empire, as Tuchanka was well and truly dead by this point, and quickly grew in power to rival their former compatriots. When they discovered the Second Core they leapt past them, and quickly used their fleets to bomb the homeworlds of all the Taiidan races in a show of force.
As a result the galactic council called in the Bentusi, and thus the stage was set for the rise of the Turians. The Palaven Hierarchy was a recently founded vassal state of the Taiidan Empire, but had rapidly made a name for themselves with their disproportionately large military. When the Hiigaran decapitation strike took out the Taiidan homeworlds it also took with it most of the traditional Asari and Salarian crewed fleets, who had never been as big as the Krogan ones anyway. And so when Bentusi dealt with the Hiigarans an up and coming Turian admiral took center stage.
Rallying the navy he swept through the Hiigaran imperial systems and took their second homeworld. He decimated their remaining defenses and then returned to the galactic council in triumph. With the Taiidan homeworlds mostly rubble little of the old leadership, including the empress, remained, and the survivors crowned the admiral as the new Emperor with a mix of awe and fear.
For the rest of his life he reigned with an iron fist. The Taiidan empire grew from a first among equals to holding almost all settled worlds in the galaxy, either directly or through vassals. Rebellions sprung up frequently, but they were brutally crushed just as quickly. Entire worlds were decimated to keep them in line.
When the emperor died he had intended for his mind to be preserved in a clone body, but the rebels saw their chance. Many of them had been alive for the entire last few centuries of turmoil, and they hadn't overthrown the Bentusi just to serve a mad emperor forever. In the unstable period after his death they revealed his many crimes. The rebellions slaughtered, the worlds burned, and worst of all the fact that he had inflicted on the Hiigarans not just exile but a terrible biological revenge.
The Taiidan Empire erupted in chaos. The emperor's cloning facilities were torn down as were most of the imperial centers of power. His cronies were killed, and a new Republic established at the will of the people. At first there was an uneasy tension between the Asari, Salarian, and Turian populations given all that had happened. But over time a balance formed and settled into a peaceful coexistence that has lasted for nearly two thousand years.
The Taiidan Republic in the modern day covers nearly ten percent of the galaxy, and the galactic council exists as a de facto extension of it. The leaders of the Republic are not afraid to throw their political weight around and most other nations bow to their whims, but they do not launch wars of conquest anymore, and over the last few centuries some of their vassals have begun agitating for independence or at least a larger role in how things are run.
Appendix: Divisions of Power
The Bentusi stay out of the affairs of the galactic council for the most part, and the younger generations are not privy to what the elders might known about the inner workings of things, but the general consensus is that the Asari are the ones running things in the Republic. Of all the surviving races in the galaxy they are the closest to the Bentusi in lifespan and it gives them a very long time to get things done.
The Turians meanwhile seem content to run the military. Very likely the threat of a coup and second emperor is what keeps the Asari from just openly taking control. Many a young Bentusi has made friends with a Turian or two on pilgrimage as the militaristic species are ever present in the Republic's navy and there are usually at least a few that aren't as stuck up on military discipline.
The Salarians remain an ever present tragedy for the Bentusi. They are so brilliant, but so short lived. If the other races lead flicker lives, then Salarians are gone in a blink. There is always a movement among the Bentusi to offer them the technology to become Unbound. And in fact the Bentusi have made that offer multiple times. But there appears to be some cultural bias that prevents them from ever accepting. In any case they seem happy with their position as the scientists and spies of the Republic.