This La Croix originates from the Third Tartarus world.
She is one of the last beings alive in the world, and she has been living in the Administrator Tower with Radio, the Cirque du Croix, and some artificially created monsters conducting research on apoptosis and chaos.
Luka and the party have ventured through the third Tartarus and entered another dimension. Within this dimension, the world itself is being consumed by chaos. Both Sylph and Gnome inform Luka that their corresponding elements are dead.
At the entrance to the tower, the party is greeted by Radio. She is currently serving as La Croix's research assistant. She fills them in on the current state of the world: it's ending. The last human died 25 years ago; the last angel died eight years ago, and the last monster died six years ago. The only building still standing is this tower. Radio offers to lead the party to La Croix, on the third floor. She asks us to be careful as La Croix has created artificial monster girls to fight off apoptosis beings, and they will likely attack Luka.
By the time the party reach the top floor of the tower, there is nothing left. Just Radio, La Croix and the Cirque du Croix.
La Croix explains that the origin point of this nothingness was Remina, and over the course of 31 years, it has consumed the world. In less than an hour, this tower will also be destroyed, and the world will be erased from existence.
The greatest minds of their world had come together in this tower to solve the mystery. Angels, monsters, and humans alike. Every day, one more person died either to chaos or apoptosis and now only La Croix is the only one still alive. But over that time, they found out the reason for all of this; it was because someone didn't follow the correct history.
The spread of chaos begun in Remina and started spreading out from there. Like a virus. But it wasn't their world which didn't follow the correct history; it was someone else's. Because of that, this virus has spread from that world to this one, and the chaos is now starting to spread to Luka's world. In time, all of the parallel worlds will be consumed, until the chaos eventually spreads into the real world, and the universe will cease to exist. Not even space. Just nothingness. She wants Luka to take what they learned back to his world for La Croix in the hope that she can finish the work she started.
There's a shudder. La Croix tells Luka to return to their world while there's still time. Luka tries to talk her into coming with them, but La Croix won't have it. She asks Luka to take Radio with them as she's grown rather fond of her and doesn't want to see her destroyed.
Luka and the party leave through the exit at the back of the room.
After they leave, Adramelech appears shortly after. She asks La Croix to move out of the way, but La Croix refuses to back down, followed by La Croix and the Cirque du Croix attacking her. You don't see the actual fight, but La Croix and the Cirque du Croix die.
When Luka encounter the La Croix of his world, the latter is seen discussing with Kagetsumugi. When her Cirque gets beaten by the hero and his companions, she tells them that struggling is futile. She changes her stance when the group gives her the note from her counterpart who died, and leaves her Cirque to be taken care of by her sister.
It's safe to assume that this world's Shirome was unable to save Chrome from the accident that turned her into a zombie. Unique dialogue taken from their exchange on the top floor of the Administrator's Tower supports this.