So at the end of Halo 4 you see chief in some hardlight bubble and in this scene Cortana and chief say goodbye, now what’s important to note is that Cortana said “most of me is down there,” and something about only a small piece of her being left to get chief off the ship. As we know the didacts ship landed on genesis or the domain and Cortana’s fragments came back together. Meanwhile chief thinks Cortana is dead due to him seeing her die. I theorize that chief saw the fragment of Cortana that saved him die, her humanity, whilst the rest was pulled into slipspace and was reassembled without it. This explains why her character ha changed, the vital part of her personality holding her back was lost. More proof? I don’t remember Cortana hating Halsey until she got to a really rampant state and in Halo 5 she still hates Halsey, this is because the part of her holding those inner grudges and desires is gone, leaving us with something that definitely not the Cortana we saw in previous games.
That’s more or less what happened. There’s actually a comic telling us about the fragments of Cortana after halo 4.
That’s certainly one of the three different explanations 343 provided to explain Cortana’s survival. Though the Dominion Splinter comic doesn’t mention Genesis at all, so we just have to wait for 343 to decide what they want.