WMG / Queen's Gambit

The Triumvirate is the Big Bad of the game overall

The biggest clues come from Tom's route, since it's the only route we see them directly. They weren't shy about hiring both the assassins they sent to kill Tom, and Shield was sent as his bodyguard primarily to test her. There's also the fact that Emily is hired on as Tom's new "aid" after Shield has to hide from basically everyone for a while. This implies the Triumvirate has contacts within the World Liberation Front since that's a group Emily is directly associated with in the other thee routes. There was also plenty of indication that the Triumvirate acts a bit like a mythological hydra - even if one (or several) heads are cut off, there are plenty more to step in. There's also the fact that these people can successfully pressure Destrier into launching an assassination attempt on the President of the US, which is against Destrier's normal operating procedure. (Destrier has a policy of no high profile targets killed on US soil, and they stay out of politics.)

Testing Shield's abilities in the field makes a bit more sense with some context taken from Emily's route: it's implied one of the reasons Emily was abandoned was not just because Emily was unruly, but Emily was also an offering to a client who had the resources and means to destroy Destrier. With her "death" she was no longer disruptive to her teammates but also, presumably, she could be reprogrammed to fit someone else's information structure. (See what happens to Shield in João's nightmare ending.) This didn't go as planned since Emily escaped. (More thoughts on that in the WMG below.)

It seems odd to Shield in João's route that Destrier, which launches hits against people who are making the world a worse place, accepts a bid from a 'close client' to assassinate João, a man who is legitimately doing good and can change the world for the better. If this client is the same person/people (the Triumvirate) from Tom's route, this makes a lot more sense. João is certainly antithetical to their practices and presents a threat to their way of doing business, they'd have every reason to see him killed. Like in Tom's route however, they're prone to bringing in outsiders (the WLF) to see the job done and again, test Shield.

They're currently the only 'client' we, the audience, have been introduced to.

Emily's been brainwashed, which is why she's so hostile in the routes that aren't her own and why her personality is so different when she isn't the love interest

In João's nightmare ending, a level of conditioning is used on Shield that alters her memory so deeply that she doesn't know who she herself is, and she certainly doesn't remember the people she used to care about. It's not hard to imagine a client that can put enough pressure on Destrier (presumably the Triumvirate) to hand Emily over would have access to personnel, equipment, and techniques needed to pull off the reconditioning that would leave Emily hostile to her former employer and colleagues, especially after such a grievous betrayal. It's also entirely possible that Emily didn't escape in routes outside of her own. Instead, she was simply... released, and let to think that her actions really were her own.