Based on the light novel series written by Tachibana Koushi and illustrated by Tsunako.

"30 years ago a large Spacequake occurred in Eurasia. 150 million people died and human life and infrastructure was laid to waste.
Spacequakes: A destructive phenomenon of unknown origin. Like the name says, it is a void which destroys everything in and around it. An unexplainable disaster. It has been thirty years after the apocalyptic Spacequake, in the six months that followed smaller scale Spacequakes occurred all over the planet. Japan, the place where I live, also faces similar occurrences. It is known as the South Kanto Void Zone. Twenty five years had passed without anything happening. But suddenly, five years ago, it started again, around the Tenguushi City area. This phenomenon increased by the day, and these occurrences seem to be centred around Japan." — Itsuka Shidou

As another Spacequake is about to hit the town, Itsuka Shidou and his class run for the shelters. Then he remembers how insistent his sister Kotori was about their meal at a family restaurant. He rushes back out into the open, when locating her GPS signal at said restaurant, frantically rushing to save her. When a Spacequake hits nearby, Shidou discovers a non-human girl, a Spirit named Yatogami Tooka, at the epicentre of the quake. Tooka asks him if he is also here to kill her. Still confused, several girls in combat suits, the Anti-Spirits Team (AST), among them his genius of a classmate Tobiichi Origami, attack Tooka. Shidou loses consciousness, awakening in sickbay, on board the Fraxinus, a floating ship located 15,000 metres above Tenguushi City.

The commander of this ship and the ongoing operation is none other than Shidou's sister Kotori! As she explains, when the Spirits drop down to Earth, this usually ends in a Spacequake occurrence. There are only two ways to stop a Spirit: one is to go on the offensive and kill it (rarely succeeds), the other, of all things, is to go on a date with the Spirit girl... and that will be Shidou's responsibility!