My take on it is mostly for our clan, and I try to leave room for things to be very different in other clans and tribes without having to contradict me
1. Also, I rather expect Matariki will have a slightly different take as well.
The Atamahara Clan belongs to the Matu'shindor sub-tribe which is a gathering of clans of similar political and traditional leanings, and maybe similar geographical origins. The sub-tribe has a bit of intermarrying for political purposes and each clan has influence according to their wealth and the prominence of their members.
The level of influence of each clan is affected by many things; One way to get more influence is to give gifts, and the richer clans are naturally more able to do this. Another way is to have prominent members out there doing things in public - Loyalist Pod pilots are therefore a valuable commodity from the clans point of view, both for their wealth and their profile.
The sub-tribe has a chief, who represents the clans under her at the next level up (I'm not entirely sure what that is, it might be a tribal council on the world) but doesn't get to decide purely on their own whim, but is influenced by the desires of the various clans and important people, with appropriate weight for how important the clan is to the sub-tribe. The Chief gets to stay the chief by having most of the clans in the sub-tribe happier with the incumbent than the trouble would be worth to change to someone else. Make too many decisions that irritate, or piss too many people off, and either some clans will leave the sub-tribe, or they'll get someone else to be the chief.
One way that it is decided how the sub-tribe will vote, or decide on a topic is that the elders will sit around and listen to representations. If you wanted the sub-clan to provide support for your pet project, you might start by talking to an elder from your clan, one-on-one, and if you can convince them they might invite you to talk to the clan council. The clan council would have a discussion after you leave, and weigh up the benefits and costs, (social as well as monetary) and if they favour it, they'll push it in the sub-clan level.
There will be trade-offs at each level. People who are owed favours, or married to someone's son, or the daughter of someone important will have their requests looked at with more favour (Kin favours kin). There will be negotiations for mutual webs of support with other elders/clans etc, and I would expect that some proposals get put to the sub-tribe so that they can be backed down on, so that the people they back down to will owe them a favour later. It's all very complicated, the rules aren't really written down, and it depends on the social interactions of the people at that level of decision making.
At the next level up, which I expect might be the continental tribal council, or the Planetary Tribal council, the same things will happen, only writ-larger. Kin ties will still be important, but so will history and competence. Charismatic types who have lots of kin ties and come from important clans will have a much better chance of getting to be a leader, or if not the leader, being influential on the council.
And finally, when we get to the Inter-Tribal council where decisions are made about the republic, every member of that council has a vast array of favours and kin and history that got them to where they are now. They each have people to reward for their support, planets that they have cousins living on, people who have pissed them off. All of that will influence decisions, but still, they will be encouraged to have the good of the republic at heart.
The way that those competing loyalties and considerations meet together would be a fascinating thing to watch, but is rather outside my level of comfort
1 when it comes to trying to describe.
In NZ, because of the treatu of Waitangi (any Kiwis here probably know more) the Maori tribes have prescibed number of representatives in parliament.
I thought Republic politics might be like that.
The Treaty of Waitangi says that there is a partnership between the Maori tribes and the Crown (currently Queen Elizabeth). One definition of this was to give mandatory seats to be elected by the Maori - Maori can choose to be on the General Roll, or the Maori Roll, which affects who they get to vote for.
Since NZ moved to Proportional representation, we now have a situation where the number of Maori seats is dependent on the population who have chosen to be on the Maori roll, so that each electorate should have a similar number of voters in it (There are some historic oddities to do with geography, but the principle is true, even if intentionally violated). The side effect of this is that if more people (of at least a certain quanta) choose to join the Maori electorate roll, then there will be more Maori MPs. If enough leave, there will be fewer.
This is basically a representative democracy, and isn't really anything like I imagine the Matari to be. That's possibly because it's normal to me, and I want the Matari to be different
1 - I can make stuff up about
my clan and my sub-tribe without worrying much about trying to write canon, but I'd rather not try to tell other people how
their clan works.