Late to Blade Runner 2049 discussion, but i really love this film and want to add my 5 cents:
It's indicated that Joi's personality is tailored through analysing the data about K she collected. Think of it as a much more sophisticated version of targeted advertisement algorithms. From that point the audience might interpret Joi's 'evolution' throughout the movie in two ways:
Joi follows into K's footsteps, and breaks away from her "programming".
Or she does exactly what she's designed for - perfectly emulates K's "gf template", based on his desires and ideas, up to her very end.
Some also argue that Joi can't have a real personality, because she's just a product. With pre-determined purpose, designed from scratch by Wallace corporation... But the same could be said about K in the beginning.
One of the things that i like about BR's sequel is that it goes deeper into Philip K. Dick's tropes of blurring the lines between "real" and not. The uncertainess is a deliberate theme that persists throughout the movie(Deckard and his dog are another example that comes to mind).
Yeah the entire premise of "is something alive if it's manufactored" is at the core of the BR franchise. And based on how the narrative, of both films, frames the characters, it's clearly meant to imply that the answer is "no, it shouldn't matter. that all life is precious and worth preserving, regardless of origin." The sanctity of life is in fact, a huge component of the original PKD story. Where all living creatures, were revered by the survivors on earth, because SO many species were dead, that any examples of a REAL animal, weren't just an oddity "Is that owl fake?" "Of course it is." but something to be protected.
I personally think it's a very moot point to try and nitpick about what an insanely advanced AI could/couldn't accomplish, in a society that has completely covered organic manufactoring to the complexity of a human being. And able to engineer them so precisely, that the idea of them having a true life/existence, becomes a key moral/ethical plot point, for 2 films. And has sparked so many offshoot stories over the decades.
Is it REALLY that hard of a stretch, that they've got sufficiently advanced AI software, to be TRULY intelligent, and thus alive? Remember, this is a society that has licked interstellar travel, and biological engineering on a MASSIVELY complex level, humans. And that was 30 years ago, in the original film's timeline. They've had decades of time to advance, and we know what the acceleration rate for technology is. So it doesn't seem odd to me, at ALL, to assume that the tech behind Joi, is sufficiently advanced, to actually be sentient, not just the simulation of it. And for that level of tech, to be so commonplace, as to be no different than us having Siri today.
From Ks perspective, though, as i interpret it: When he sees the giant Joi hologram he's at his lowest point, after just learning that he himself is not special. At this moment, whatever real remaining from his relationship with Joi, died for him. That coaxes K into going after something more tangible, as he decides to rescue Deckard - symbolically first and last actual decision in his life he makes, IMHO.
I don't think the relationship "died", so much as he was just confronted with an open, bleeding wound. I suspect he was probably, on some level, tempted to answer her call of "you look lonely....I can fix that." Because she DID fix that for him, and so much more. And I'm sure a part of him was screaming "You could have her again!" But, BECAUSE he truly felt her to be real, a statement he makes to her during the sex scene "You ARE real to me." He couldn't bring himself to do it, because he knew it wouldn't be "his" Joi. Not really. But like anyone who has had a relationship end due to death, it's still with him, the emotions, the feelings, the significance of it.
BUT, because Joi also (my theory anyway, given PKD's overall setting tone) is basically an emotional prosthetic, to try and help people reconnect with society at large, he had the emotional stability, AND a personal connection, to several people at that point, to allow him to move on, not become a recluse in his apartment again, and try and help others. It helped to stimulate his empathy.
