From MangaHelpers:
This manga is related to the "Date A Live" novel and anime. It features members of the AST (AntiSpirit Team), a special squad devoted to fighting spirits. Okamine Mikie idolizes her superior AST member, Tobiichi Origami, and may be in love with her as well. Unfortunately, Origami seems to want nothing to do with her. One day, they end up heading toward an explosion that wasn't caused by a spirit. Instead, it was caused by a girl named Ashley Sinclair, who transfers to their school shortly after the incident!

Did you know there was an organization that is protecting the world from the harm? Its name is Ratatoskr.

Their job is to assist "Spirits", an unknown beings that causes destruction to wherever they go, whether they intend to or not. And they help those "spirits" adjust into normal life.

This is a story of a boy, Itsuka Shidoh, and his daily life, where he was forced into attempts into leading the spirits into his regular ordinary way of life and where they lead... his?

April 10th. Yesterday was the last day of spring vacation, so starting this morning it was a school day. After being woken up by his cute little sister, Itsuka Shidou believed that it would be the start of another normal day. As for predicting the chance meeting with the girl who called herself a spirit...... Along with a sudden shockwave, the townscape vanished without a trace. In a corner of the street which had now become a crater, the girl was there. "�”You, have you come to kill me too?" She is the disaster that will destroy humanity, a monster of unknown origin, and a being rejected by the world. There are only two ways to stop this girl: annihilation, and conversation. His little sister Kotori, wrapped in a military uniform, thus said to Shidou: "Since it's like this, just go date her, and make the spirit fall for you!" "Wh.. whaaaaaaaat!?" Enter a new age of boy meets girl!!

4-koma Based on a Novel - Date A Live.