Domestic Avengers is a relationship between the Avengers team, often incorporating S.H.I.E.L.D. characters such as Phil Coulson, or even villains like Loki.
The Avengers all met each other after SHIELD gather them after Loki steals the Teseract. They defeat Loki and later go out for Shwarma, before going their own separate ways.
Age of Ultron
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The style of fic that the Domestic Avengers would be a part of, began to spring up after the release of The Avengers. Many write that Stark Tower had become Avengers Tower, a theory that later became canon, and that all the Avengers were living there. Each Avenger had their own floor, or space but there would also be a gym and a group family/entertainment room that they would spend time in together. Many fics in this style would also typically have the author's preferred pairing as the focus, with the Domestic Avengers as a setting or background aspect.