I have worked with magnets for over thirty years, yet I admit that I'm not sure if magnetic fields are real.
Now, they tell you that the magnetic field is real, real like the river's undertow that pulls you past the eddies beneath the sharp rocks and beats you down, but it's not. Not really. You feel magnetic forces, to be sure, but the magnetic field is not the fundamental truth. The truth is the magnetic vector potential. And this is not just my opinion; this has been known for over sixty years.
The magnetic field (what you hear about) is written as B; the magnetic vector potential (what you perhaps never heard about before today) is written as A. They are related by:
B = ∇ x A
("The magnetic field is the curl of the magnetic vector potential.")
The truth is revealed by the Aharonov–Bohm effect, in which particle traveling past a long thin solenoid experiences a phase shift, even though the magnetic field is confined to the interior of the solenoid, which the particle never enters! Why does this happen? It happens because the particle sees the magnetic vector potential outside the solenoid. (A cool animation is available.)
So, A (the magnetic vector potential) is what's true and real and actually felt by the particle, while B (the magnetic field) is sort of an emanation, or a mathematical device, that almost always gives you the right answer (except in clever made-up configurations designed to demonstrate the Aharonov–Bohm effect, that is), and is handy. I suppose an attorney might say that while the magnetic field is (usually) a truth, it is not the whole truth.
Perhaps it's like your car's engine, where horsepower is what you read about, but torque is what you feel. The magnetic field is what you read about, but the magnetic vector potential is what you feel.