What's happened in the real world?
The real world aspect of the game is set sometime in the near future from present day.
The backstory is that a woman came to prominence, who is generally known only by the title of The Prophet. Her aim was to bring about world peace and harmony - and she's being very successful in this.
The game starts during her accomplishment of world peace and harmony - it's well on the way, but not totally there yet. People in the real world are, however, sure that it will be achieved within their lifetime.
Most of the world's major war zones have stopped fighting and put down their weapons and there are moves towards peace and reconciliation. It's all going remarkably smoothly - or so it seems on the surface. In day to day life, people are getting on better. Stress seems to be becoming a thing of the past, there are fewer arguments, everyone is just getting along, there's no longer the push and shove of getting ahead in life no matter what it costs. The world is just becoming one big, fuzzy, happy love in.
Not everyone believes in the Prophet's message, though those who do not are few and far between. The most prominent of those who don't is known as the Betrayer - a former follower of the Prophet who foresaw that true peace and harmony would cause the end of the world. It was the Betrayer who caused the Wakings to fall into comas, meaning the creation of Eidolon City. The Betrayer continues to try and find a way to defeat the Prophet.