So the story goes that their relationship was that of Zato-1 being obsessive/abusive and possessive of her, and although her feelings for him were that of love, this didn't last long. After the God-Soldier series events (Night of Knives), her desire to live outweighed any orders from the Guild...therefore the next mission she was given by them which just so happened to be a mission in which Zato-1 himself was involved, she sold him and his subordinates on that mission out to the IPF (International Police Force).
Zato-1 was imprisoned and his Kinjuu sealed to dormancy. It wasn't until the events prior to the 2nd Sacred Order of Holy Knights Recruitment Tournament (Guilty Gear: The Missing Link) that Testament set Zato-1 free, informing him that the spillage of blood was allowed for this tournament (for the purposes of breaking Justice's blood seal).
It was during that tournament that PostWar Administration Bureau managed to keep an eye on Zato-1's movements but were unable to capture him, though they did manage to obtain Ky Kiske's initial combat data for their RK series.
Zato-1 managed to finally meet with Millia, but the meeting was short-lived as not only did she manage to defeat him, she fled him, declaring full independence from the Guild. Though he wasn't seriously wounded and managed to even fight the likes of Justice at one point, his power-drunkenness awoke the Kinjuu that had been incubating inside of him.
The birth cries of the Kinjuu (Forbidden Beast) itself were what ultimately killed him as it took over every aspect and function of his body, his mind and body could not take the stress. The Kinjuu realizes it has killed its host and names itself Eddie. Realizing it can no longer survive without a host (and also realizing it's given lifespan limitations programmed internally into it along with the predisposition to hunt gears as its natural nemesis), it sets out to find a new host. Dr. Faust has been monitoring him for some time as well.
It was shortly thereafter that Venom sets out to find Zato-1 or what remains of him, and during the Gear bounty events (Guilty Gear X: By Your Side G.Gear) he ultimately confirms Zato-1's "death".
This is also where Millia makes an attempt to sever all ties to Zato-1, including Eddie himself.
It's currently unclear the whereabouts of Eddie or if it died or evolved past the need for a Host or was captured and put in a test tube for study either by the PWAB or That Man...either way we don't know. For a time Venom pursued it in attempt to recover the body at least but that has failed and so he himself tasks himself with the destruction of the Kinjuu (and perhaps Millia's Kinjuu should it come to it).
It seems that Venom has been given reign over the Guild with Slayer's blessing, though Venom holds an equally strong grudge against Millia, it's not certain if he will actually pursue her. Slayer has given Millia fair warning that her Kinjuu "Angra" is on the brink of "being born" and what this will do to her body, but we can't say for sure what will happen or has happened during or prior to GG2: Overture events, only that it's possible PWAB is pursuing her, most likely to recover the Kinjuu they implanted in her body (like Zato-1 was given initially).