Say that Desmond survived his electrocution - instead of killing him, it just put him in a coma for a while. When he wakes up, the Order insists on using the Apple's power to wipe out a few Templars, and see what else it can do. Desmond, too tired to argue and just wanting his life to be over by now, agrees and uses the Apple to kill the Templars (your choice here) before he tries to use it for something else.

That "something else" turns out to be granting a personal wish that Desmond had when he was younger (and still has now, even though he won't admit it) - finding someone who can actually love and take care of him. Not as a friend, or a lover, but as a PARENT. Cue the arrival of Giovanni Auditore, who after seeing the hells Desmond has to live through on a daily basis now, takes the broken boy under his wing as his adopted son and starts to train and raise him like he did Federico, Ezio and Petruccio.

William does not take kindly to someone trying to snatch his sacrificial lamb, er, I mean SON away. Neither does the Order, for that matter.