Which one is right, are EM fields made of photons or are they fundamantal?
A classical "electromagnetic field" cannot be defined in the way an electric field or a magnetic field is defined classically (or a gravitational one). One needs a test particle to measure the strength of the field, and I cannot define a test particle that will measure the strength of electromagnetic radiation. One can define an electric field and magnetic field at a point in space-time , and Maxwell's equations connect them as one entity that behaves differently given the velocity of the observer.Two different kinds of test particles are necessary to test electric and magnetic fields, and that is part of the confusion.
To start with, since from our present knowledge, everything classical emerges from the underlying quantum mechanical level, fundamental are the photon particles represented in quantum field theory by a photon field, on which creation and annihilation operators generate the real photons.
Both classical Maxwell equations and the quantum mechanical ones are mathematical models. It has been demonstrated mathematically that the classical fields emerge from the quantum mechanical ones, and there is smooth continuity, going from particles ( photons) to electromagnetic waves(classical ). It can also be shown that at the limiting case of static behavior this continuity exists, and virtual photons can mathematically model static fields.
As physicists we accept what the rigorous mathematical models predict and describe, as long as there is no experimental falsification. So at this point in time, fundamental is the photon field with its creation and annihilation operators ( an operator field). Virtual photons are a price for using mathematics, as they cannot be measured, but as far as the theory goes, the whole thing hangs together with no experimental falsifications.