From a certain point on, it is extremely difficult to discriminate between "real things" and "non-real things", as you defined these terms.
You see, the mathematical abstraction of Hilbert spaces may not be something that you can feel, but it is surely something that can be used to determine "real" things. But these topics are way too philosophical for them to be an answer that satisfies you, I believe.
We use mathematical abstraction in physics in order to predict the behaviour of "real" objects, like light. Forget that some theories are still in development, for science always progresses over time. This abstraction, however, has to be used in such a way that we're trying to assume that for every rule or event that happens in physics, there is some mathematical theory that has the same axioms and can be used to determine a result beforehand.
This may not always be true, and mathematics can only offer a (somewhat elegant) approximation of physics. It is not true, in fact, that energy varies continuously over time. But you surely can build a lot of stuff and solve a lot of problems by assuming it does, and apply theorems for continuous functions in your physical calculation.
Now, on to fields. A field is a mathematical abstraction. There are scalar fields, vector fields, etc. Still, they are there, they are real: we use this abstraction to predict the behaviour of an object that really is there. If you had an organ that reacts to downward forces, you would be able to tell the variation of Earth's gravitational field. Similarly, many animals possess organs that let them feel the electrical field around them changing.
This being said, you do not need new organs to discover something human body can't see, hear, smell, taste or touch. You're just going to "convert the information into a format you can read", so to speak. You can see light, but you're never going to measure the speed of light with your bare eyes; that is the essence of experiments.
Mind you, mathematical abstraction are often so closely related to "real" stuff that they're confused with each other. A vector is not real, a force is. But we're so used to vector forces, we swap them carelessly. We can do so only because we know forces behave (with an extremely fine approximation) as vectors, but vectors are not real.
Vector fields are not real, for the same reasons vectors are not real; the electric field is real and you could feel it if you had the right organs. Hilbert spaces are not real, superposition is.