Rust is a game developed by Facepunch Studios, which was heavily inspired by Day Z. It was released in early access in 2013, and fully released in 2018.
While Rust is also an open-world survival game, the threat comes almost solely from other players instead of zombies. The survival elements are similar, with players required to eat, drink, and tend to wounds, but the addition of a crafting system made the game stand apart from Day Z. Players can craft weapons, clothing, shelters, and other game elements to give them an advantage over the environment and other players.
Since Fallout 4 introduced the concept of shelter building in the Fallout franchise, it's likely that Bethesda developers pulled from this game for inspiration on how to make that work in a multiplayer setting.
However, Rust's final release received only average reviews, and by 2018 it seemed that consumers had become tired of the open-world survival genre.