He's not alive. The Desmond we all know is completely dead, gone, deceased... however you want to call it. Abstergo took some of his blood and organs to create the Sample 17 Project. The Desmond that "lives" in the Gray along with Clay is more like an Artificial Intelligence that has Desmond's memories and personality. Just like Clay in Revelations.
Desmond Miles’ face is on every security screen in Abstergo in black flag - well after his ‘death’ is officially recorded. If I had him in storage and didn’t want him busted out by assassins, I’d make sure my ‘private records’ labelled him dead and referenced his autopsy too.