the way the Bach Cello Suites exist, or the way Harry Potter exists. They exist conceptualized in someone's head. They might be logically consistent, perhaps even computationally useful, but whether they have any referent in the "real world" is something else again.

For example, the acceleration vector fields described by classical gravitation and Maxwell's E&M equations certainly have a mathematical existence and application, but in General Relativity it has no use or reality at all. The Newtonian force fields are merely a mathematical abstraction for the behavior of an Einsteinian geometrical concept under certain constraints. It's all in our heads.

The "force field" does not exist any more than "action at a distance" ever existed. Each is a conceptual shorthand for a reality more fully described at some other level.