Shadow of Memories/WMG

The entire game is a hallucination Eike is having as he lies dying.
Hey, it makes about as much sense as anything actually in the game... Although it doesn't explain the multiple endings. == Homunculus continued to mess with Wagner/Eike (such as mind wiping Wagner every few years, or even transporting him throughout time) during the 400 year period between his memory loss and his first death. == It explains why Eike doesn't remember anything that happened during the 400 year gap. And Eike's distinct lack of any family or friends.

The backstory revealed in Ending D is true, and the Extra endings are real endings.
When Dr. Wagner released Homunculus, his soul was bound to return to Homunculus when he dies. Wagner then sealed up the Homunculus back into the stone. Mind-wiped Wagner wandered outside the city with this stone for hundreds of years. Eventually he comes back, before the beginning of the game, as Eike. For some reason, he forgets he's had the stone the whole time, and loses it in the cafÃ©.

Thus, Homunculus is trapped in a time loop where he is created, sealed up, lost in a cafÃ©, given back to Eike, given to Dr. Wagner, created, sealed up, and so on. However, Dr. Wagner's soul is still bound to Homunculus. So when Eike finally dies after the game's events (as in Ending C, or some future occurrence), he's brought to Homunculus. However, Homunculus only exists in a time loop, so this future Eike must be brought to a point in this time loop. In this case, that's the start of the game.

This is where the Extra Prologue starts. This Eike is from after the end of the game, so he remembers everything that happened in the game. This allows him to take the proper steps to either stop Homunculus from being created (telling Wagner to make the elixir) or removing him from time altogether (throwing the stone at him).

Sure, it's not perfect (I still haven't figured out when Homunculus has the time to be out and about and switching babies), and it wouldn't explain theoretical future Eike replacing original game Eike during the Extra Prologue, but I think this is pretty close to a "real" ending.
 * Makes sense. I don't know where people keep getting the idea that the Wagner thing is a seperate/alternate timeline from (unless it was Word of God). Personally I've always seen each ending as possible, given Homunculus explained that for every action in timeline, another, different action occurs depending on the choices made. So in one timeline, Eike fixed everything (returned the right girsl to their times, stopped his murderer, and defeated homunculus), while in others, he died, got piss drunk, doesn't solve anything and only prevents his death.