I'm not sure that this is what you are looking for, but here is a story you might find interesting and relevant.
Charged particle are feeling forces generated by two fields - the electric field and the magnetic field. And already in the 19th century people knew how to write the equations that describe the motion of these particles due to these fields.
These two fields are in fact related, and it was discovered that you can derive them from a single entity that is called "vector potential", which unifies them. However, this vector potential is a peculiar thing, it turns out that there are different ways to write it and give it different values, while the magnetic and electric fields that are derived from it remain that same! This freedom is called "gauge" and we are saying that the theory of electromagnetism is "gauge invariant" - no matter what specific gauge you choose for the vector potential, the physical results are the same.
This will make it seem like the vector potential is an invented quantity. Some mathematical side-tool that allows us to treat both the magnetic and electric fields - the "true" entities - in a more convenient way. But it turns out that when you try to write the quantum description of the electromagnetic theory, this vector potential is exactly the field that describes it. In a sense, this is the "true entity" the the magnetic and electric fields are just its manifestations.
The gauge symmetry remains - you can still write it in different ways. But now, it turns out that this symmetry is much more than a mathematical trivia point. It is a fundamental symmetry of nature. In fact, when we write theories that describe more complicated phenomena (let's say the nuclear force) we now start with requiring that the force will described by something like the vector potential, which has a gauge symmetry, and the nature of the gauge symmetry tells us almost all we need to know about the force itself!