At work now, but in summation:
The Amazons use a magical system vaguely similar to Hermetic Magic, due to Viea's influence. It is more limited, however.
The Augustans, from my brief survey, use a different system not founded on existant rules systems, but not as comprehensive as a Hermetic System. More on this as I pick it apart.
The Muspelli predominantly use Supernatural Abilities, but use their own ritual magic-style system to empower them beyond the bonds of a common user of Hex and so on. They also have a new system of "empowered form" adoption and advancement.
The Soqotrans use a magical system founded around being bound to spirit beings while young, with education into a magical tradition/focus based on your tribe (Olibanum, Aloe, Cinnabar, Myrrh, and...Robot. No... ok, went back and looked. There's no 5th tribe.). ALSO, their magic is based on their 4 magical skills, which are Supernatural Abilities.
Aloe: Restoration/Regrowth. Want to make a sword from a sliver of a sword? Aloe-a-go-go!
Cinnabar: Preservation. Even of things like luck, or courage.
Myrrh: Commemoration and enchancement of a thing's "mythicness". Want to be as farsighted as Argus? Apply here.
Olibanum: Magical commands. As noted in the text, this is the most readily equatable to a Hermetic Art. One Confidence point if you can tell me which one...
The spirit allies of the sorcerors are linked to magical trees and plants on the island, by the way, thus the names of the tribes and magical traditions. These spirits provide instruction, so a sorcerer likely needs to stick close to home if they want to learn things. Spirits are generally not gung-ho about leaving the aloe patch they are connected to untended, so the Soqotrans tend to stick close to home.
Soqotrans are pretty awesome in their homeland - which they rule, but not necessarily outside of it. A fair chunk of their grimoires (though by no means all of them - there is a fair spread of interesting effects) are tailored to interact with their somewhat exotic and fantastical island chain, the spirits thereon, and the things sorcerers in their position care about: trade, ships, dealing with hostile spirits and so on that an island-bound group of sorcerers need. Magical geckos, giant centipedes, magical herons, jinniyah good and bad, and so on are also described and statted out. These are important as service to the spirits and the island can equate to enhanced assistance and power from these spirits. Might points can be used to boost your magical die rolls, for example...
They've got some interesting Range/Duration/Targets, based on censer, incense, and similar.
They also NEED to spend vis to cast their spells, at minimum 1/10 of a pawn.
I'll be giving them a fuller look later. They look like a lot of fun, and I am saddened that my troupe just left the East after a year adventuring, questing, and so on. Soqotra would have been a lot of fun!
My brief thoughts on Rival Magic:
If you liked Ancient Magic, Rival Magic is a pretty keen book.
If you wanted to have some interesting antagonists or foils, Rival Magic is a pretty useful tome.
If you are interested in Order Politics, Rival Magic has some neat options for "foreign powers" for the Order to engage with in various ways. The Muspelli section also provides a "Norse Magical Compact" featuring pretty much every Norse magical tradition (or some extrapolated ones from Hedge Magic) we've seen in 5th edition joining forces into a rough sketch of a "Norse Magical Society" that a saga could potentially be constructed upon. Not the "Order of Odin", but probably easily misinterpretated as them.