So the main point of this thread is trying to find ways to incentivize people to play the theme of the divide and at the same time find it in their best interest to do so, so that they RP it voluntarily and not just for show.
What things could we add to the game that could achieve this objective?
For example, what would/could motivate a Mixer to hate corpies and at the same time benefit him from doing so?
I am going to build on what Grey0 suggested (basically a numerical faction system), because I also came up with a similar idea. Which I agree might not be best but goes into the scope of possible incentives and deterrents (And people are going to hate it, because nobody wants to lose stuff).
A Mixer could have 'Mix Street credit', and they could improve that by doing an X number of actions related to what the staff thinks its themely for a Mixer to do (kill corpies, do crimes, Defy the law, help gangs, etc).
Let’s say there would be some kind of automated system that keeps score of that and then you can also sporadically be awarded by GMs boosts (x2 points) or bonus points if they noticed that you are 'staying in character' and RPing that themely Mixer consistently or it could also be in reverse, you lose points or are given a debuff by doing actions that would be unthemly for a Mixer (Being spotted eating Sushi Topside, Obeying the Law, Running Boxes in Green, Helping the corporate factions, etc.)
Depending on your 'Street cred', certain 'standard' services or NPCs would be available or denied to you and if you have plenty of points, you'd get access to 'special' services and NPCs of the Mixer faction.
This would reward people who stay themely and hinder those who aren't. Your street cred isn't good enough? Can't join a gang, can't get a lease, can't enter X hangout or get access to X NPC (Ripper Doc, Gang Leader, whatever).
You have a legendary street cred like Seven Ecks? Mixers trust you, they want to help you, and they might even follow you. You maybe get free drinks, or discounts, access to special shops, or you can even recruit random generic Mixer NPCs to follow you and ordered them around, etc.
So basically you could earn cool stuff that is both awesome and themely in accord to your faction.
What about corpies? They get something similar, you are loyal you get access to Space, Aerodyns, Biomods, even MORE perks, etc. You are an exemplary corporate citizen? Then you get access to exclusive hangouts, items, even MORE RIDICULOUS perks, corpie personalities (Meet Chen Mayo or Juicy V) and maybe you even get to be a CEO for a bit, while your boss vacations in Jupiter.
So yeah, this would sort of force the players to work for what they want to have and invest in either side of the divide, you work for one and you lose face with the other.
Corpies would have to actually work for their careers, instead of being handed everything the second their work terminal says they are 'corporate' and get access to anything they can afford, which doesn't take much effort earn in my personal opinion.
A system like this one creates scenarios where money alone isn't good enough and you also need to have a reputation (based on IC actions and RP) too in order to back it up.
This would also generate 'faction pride', players who managed to get high enough in their factions REP hierarchy on either side would have certain renown based on their actions, RP and the exclusive goodies they gained and most likely would chastise or encourage members of the same faction to improve their REP.
No faction standing? Never heard of you, you aren't important enough, sorry. But you get to move around freely and with less scrutiny.
You have negative faction standing? Then you get drawbacks whenever interacting with your opposite faction and most likely will get threatened or even harmed if you make the trip to the other side of the pond.
Anyways, keep the ball rolling there are already some pretty good ideas in here.