FWIW, I think of mathematical models of quantum field theory as like to an experiment rather as a map is like to the landscape to which the map acts as a guide. It takes experience to learn how to use a map that is much less than isomorphically like to the landscape, with some things perhaps rendered almost photorealistically but with other things located on the map only as a symbol, with much variation of style from one map to another.
What the mathematics of QFT says depends on your choice of axioms. Almost universally, there are states over an algebra of operators that is freely constructed using an operator-valued distribution ##\hat F(x)##. The operators are of the form ##\hat F_{\!f}=\int\hat F(x)f(x)\mathrm{d}^4x## (where the "test function" ##f(x)## constructs an average of the operator-valued distribution that is differently weighted in different regions of space-time), and satisfy commutation relations ##[\hat F_{\!f},\hat F_{\!g}]=(f^*,g)-(g^*,f)##. The standard vacuum state of the free field is $$\langle 0|\mathrm{e}^{\mathrm{i}\lambda\hat F_{\!f}}|0\rangle=\mathrm{e}^{-\lambda^2(f^*,f)/2},$$ which, if ##f=f^*## is real, is the characteristic function of a normal distribution. The inner product ##(f,g)## fixes both the variance of a normal distribution associated with any given operator ##\hat F_{\!f}## and also the commutation relations that determines where Bell inequalities are violated, where Wigner functions take negative values, et cetera. The form of the inner product makes no difference to the algebraic structure, however, only to the geometrical structure. For interacting fields the characteristic function ##\langle 0|\mathrm{e}^{\mathrm{i}\lambda\hat F_{\!f}}|0\rangle## must be deformed to correspond to a different probability distribution, but still there are, in principle, the field operators ##\hat F_{\!f}##. [All the above will seem quite distant to anyone who has learned only the path integral approach and works with time-ordered products of operators, but such constructions can be regarded as derived from the construction above.]
How does this structure relate to an experiment? The best approach, again FWIW, is to work with QFT as a signal analysis formalism, where a test function is called a "window function" (I've also seen Chris Fewster call it a "sampling function"), particularly because all the experimental raw data certainly comes to us as voltages on signal lines (think of CERN, with many thousands of measurements of voltages on signal lines being compressed by hardware and software into times and geometries of many events per second, which are then grouped together so that statistics can be compared to the probability distributions that a given theory proposes; but even a very small experiment does the same kind of signal analysis for far fewer voltages on signal lines).
To get back to the original question, the voltages on signal lines can be said to be "real", if you like, although it makes little difference to the world whether you call it names or not, then we might call the operators that are used to describe the signal analysis we do of the records we have of the voltages on the signal lines as not "real" in quite the same sense, because their relationship to the computing and experimental hardware is somewhat more remote (but this is just to say that we have in this a different kind of map of the experiment, which uses different symbols, a little more like an electron microscope and less like a telescope for the sophistication and directness of the transforms that are being applied, say.)
Let me add that a signal analysis approach allows us to say of states other than the vacuum state that they are modulations of the vacuum state. There is a difference from ordinary field theories, because here the modulation is a modulation of probability distributions associated with a signal line voltage, not of a single value for a signal line voltage. Finally, classical signal analysis is very much about Hilbert spaces, because Fourier transforms are very naturally associated with Hilbert spaces, and classical time-frequency analysis is very much used to working with Wigner functions, so that QFT is very close indeed to classical signal analysis.
I imagine no-one will read to here, but to anyone who does, good luck with your own engagement with QFT.
, although a four-dimensional version of classical signal analysis, which I am won't to call a random field when it is generalized to a probabilistic formalism and presented in a Hilbert space formalism, is rather close to QFT. Close enough that one can in some cases construct isomorphisms between Hilbert spaces, and, with more construction required, between the respective algebras of operators of the quantum and random fields.