Complete tasks to aid the Arcanist cause.

The Arcanists are one of three factions in Anthem. To gain loyalty with a faction, you must talk with their personnel in Fort Tarsis, complete contracts they request, and get collectibles related to them. In addition to reaching Loyalty 3, you need to completely fill the loyalty bar (essentially reaching a non-existent Loyalty 4). You shouldn't need to go out of your way for any of the loyalty achievements, but if you have completed the story, all collectibles, and gotten your Masterwork rarity, you can do additional contracts from the board next to Matthias in the building next to the fountain and open treasure chests while on missions for the Arcanists.