Sephiroth (the real 'alive' one) died in that reactor. Now through out the whole story, the appearances of himself that we have witnessed, I believe that it's obvious that Sephiroth was a ghost after all. Basically, he became a ghost after his death.
Although he was being infused with Jenova cells, that's how he has managed to gain power whilst he was a ghost, he was trying to get more power (Black Materia, etc) so his powers will make him more real, in which case, did happen, sort of anyway but when Cloud killed him again, his ghost simply disappeared. The source of his ghost (Jenova cells) still exist, that's why he appeared in AC again.
[Aeris's Ghost in that Chruch during FF7)
[Zack's Ghost in the photo of AC]
[Zack and Aeris walking and fading away from the end of AC]
Even if you still don't believe that Sephiroth became a ghost after his death, this still prove that there's really an existence of ghosts in the FF7 unverise.