The Turian Victus was given the title of "Primarch" after the previous Primarch died. Apparently this required some complicated decision making from the a Turian council.
What rank is Primarch exactly? What power does it hold and how is it decided? Is it analogous to any current political or military position?
The Turian society is highly militarized and based on a meritocracy. Throughout their lives, Turians are promoted and demoted throughout 27 "citizenship tiers" by proving their worth and value to their society. Primarch is the highest rank you can attain.
The Codex entry on Turian government says the following:
At the top are the Primarchs, who each rule a colonization cluster. The Primarchs vote on matters of national import. They otherwise maintain a "hands-off" policy, trusting the citizens on each level below them to do their jobs competently.
I'm finding difficulty with an analogue to existing governments because I'm not sure how large the scope of "colonization cluster" means, whether it's small colonies on the planet itself or planetside colonies as a whole.
Considering the import of how necessary it was for them to figure out the next in line of succession, and that Primarch Victus is able to speak for the Turians during the meeting between the Krogans and Salarians, I'm leaning towards the latter, in which case I don't think we currently have an exact analogue of someone who speaks for the entire planet.
However, if you think of the separate colonies of the Turians as states, then you could roughly think of the Primarchs as Senators (in regards to the U.S. government) who vote on matters of import for the colonies/planets they represent. This analogy is a little flawed though because given that Palaven is the Turian homeworld, the Primach seems to hold a bit more sway in that he's allowed to represent the race as a whole.
In short, there's not a directly analogous position given the scale of power, but we can approximate.
From what I have observed I'd dare say that a Primarch's equal on a earth basis would be President. This is not an exact equal though given our current government system. I believe it would be an exact equal though if there was a president of Earth as a whole. So for instance say we have a President of Earth and then President of the moon. The exact equal for the Turians would be a Primarch of the Earth and then one for the moon. This is simply a theory though. I have some issues with my theory though. One such issue would be how easily a Primarch was appointed. On Earth Presidents are obviously not appointed so easily. I assume the reason it went so easily though on the Palaven moon was because it is a different form of government entirely. Another issue I have with my theory is how easily this Primarch just left the planet, moon. This issue leads me to think that user68722 may be closer to the truth than my own. But again I even have a issue with that reply, mostly being how easily he just left. So with some thought I think there may not be a exact equal to earth's systems. But hopefully I raised some things to think about before a final answer is selected.
The role of Pirmarch fills the same role as the Princeps or First citizen. In the Early Roman empire where the Emperor/First Citizen still shared power with the Roman Senate.
Because the turians are a hierarchy i would say that they are more akin to a monarch, however if we try to put an Earth example, the closest thing would probably be Ban Ki-Moon the UN Secretary-General as He is the only person with authority to speak on the world's behalf because of the position.
