Per the ARG book, both Coryan AND Altheria use red cloaks to denote their military. Since 1) Altheria did not leave the Empire as violently as the rest, 2) maintains ties to the Mother Church, and 3) there are many Coryani serving in the Shining Patrol, I am sure they wouldn't mind seeing Altherians in that colour on the street
[Cody's completely unofficial over-analysis of Arcanis culture time!]
To the concept of clothes and colour, ancient peoples in our world were VERY tied into the colours they wore. Dyes were expensive, and so were certain fabrics, but moreso the clothes you wore were your identification in society. People who wore clothes "above their station" (like a commoner wearing purple, a royal colour) were considered to be liars or charlatans because they were publically proclaiming themselves to be something they weren't. Two good examples of this are the Toga, which was an Etruscan and later Roman way of separating themselves from their neighbouring cultures, and the ponchos worn in the Inca Empire which proclaimed what region you came from. Remember, this is before photo ID was a thing, so what you wore was the equivalent of having a passport.
In Arcanis, there are a few markers of this. As stated, the Toga is a mark of Coryani, and specificially Coryani citizenship. The coloured stripe then would denote what grade of citizen you are, with the gold representing Patrician Imperialis, the highest. The Coryani breakaway states of Altheria, Canceri, and Milandir do not use this system at all, but have other social ways of reinforcing social status within their nation (such as the Milandisian Pact of Oaths and the tabards worn, likely bearing some device to identify loyalty to a feudal lord).
Colour gets more complicated in Arcanis because of the strong association of colours with religion, rather than simple "purple dye is expensive so it is a mark of wealth." Different colours would be associated (most likely) with the different aspects of the Gods, and if we follow that than white is the "royal colour" among humanity because of the association with Illiir, while purple could be the colour of law and magic and the like because of the association with Sarish. We see a lot of pictures of Coryani patricians wearing snowy-white togas and tunics, which could be a purposeful (though likely an incidental) representation of this. That said, there have been references in Arcanis lore which contradict this, such as stating that black is a colour of mourning even though the two Gods most associated with it are Neroth (brown) and Beltine (yellow).
Overall, it not perfect system.