Is the electromagnetic field a real physical entity? Up till now, I always thought that electric and magnetic fields are mathematical constructs which aid our understanding. What was a one-step process of particle $A$ exerting force on a particle $B$ is replaced by a two-step process of the particle $A$ creating a field and the particle $B$ then entering it and experiencing the force.
But I just read a chapter on electromagnetic (EM) waves. From what I understood, if a charged particle $A$ moves, the electromagnetic field induced by it also changes, and this change in the field can be modeled by a wave equation. This wave equation would then determine the force experienced by a particle $B$ when it enters the field at a particular position $\vec{r}$ and time $t$.
My confusion starts when I see that these varying EM fields (or EM waves) produce an actual physical quantity called light. So if EM Fields were imaginary, then these EM waves should also be imaginary. But expiriment and expirience shows that they are very real. Hence the only possible reason for my confusion is that EM fields are real-world entities. Can someone confirm this for me?
 A: You should consider electromagnetic fields to be just as “real” as matter because both have energy, momentum, and angular momentum.
“Reality” is a vague concept and isn’t what is important here. What is important is that energy, momentum, and angular momentum can only be locally conserved if the EM field transports them.
In the Standard Model of particle physics, everything consists of just seventeen fields. For example, there is an electron-positron field. So either all seventeen are “real” or none of them are. It makes no sense to say that the electron-positron field is real but the EM field isn’t.
A: You first correctly learned that a classical EM field was a way for a charged object A to effect a charged particle B that was some distance away (aka action-at-a-distance).  It does seem like an "imaginary" mathematical construct.  But this "imaginary" field must carry energy/momentum because the energy/momentum of A and B change during the interaction.  So the field is behaving more like real stuff.
Finally, quantum mechanics arrives and says an EM wave can be treated as a particle (the photon) which is as real as any other particle.  The photoelectric effect experimentally shows that an electromagnetic wave delivers packets with particular chunks of energy (ie: photons).  I think your amazement that EM waves are "real" is just the amazement everyone had in 1905 about Einstein's quantum mechanical explanation!
This wave/particle duality exists for all the force carrying particles:
1) EM wave (an operator Q which doesn't change the charge)=photon particle
2) Weak Interaction wave (SU(2) operators which raise/leave unchanged/lower weak isospin= W+, Z0, W- particles.
3) Strong Interaction wave ( SU(3) operators which have 8 actions on the 3 colors a quark has)= 8 gluons.
Actually Weak and Strong interactions were never treated as action-at-a-distance waves, but they were thought of as point interactions. The SU(2) and SU(3) operators did their thing to A and B, but only when A and B overlapped in space.  The range of the force is inversely proportional to the mass of the force particle, so the EM range is infinite and could be treated as a wave because the photon mass is zero. The range of the weak force is very short because the W and Z bosons are massive.  The range of the strong force is infinite because the gluons are massless, but we don't experience it every day like EM because there are no stable particles with color. When one attempts to get color by pulling apart a colorless particle made of colored quarks, the potential energy put into the spring-like strong force creates q-qbar pairs from the vacuum.  These "cloak" (=bind to) the colored pieces and only colorless stable particles fly out.
Your amazement that the classical fields A uses to do things to B (whether long or short range) are also particles is at the heart of modern quantum mechanics today.
A: In general the magnetic and electric field arises from the more fundamental electromagnetic potential and the Aharonov-Bohm-Effect shows that this potential has a real physical meaning rather than being a purly mathematical construction. In mordern physics both fields are combined to the electromagnetic field which can be described as the curvature of a g- valued connection 1-form on the g-principle bundle of a 3+1 dimensional riemannian or lorentzian manifold; the electromagnetic potential or gauge potential. If the connection associated with a point in space and time is curved, a fieldstrength can be measured. But this is only the theoretical description of a very real phenomenon. An electromagnetic wave is only the manifestation of time dependend electric and magnetic fields creating each other according to maxwells equation. In QED photons are gauge bosons which mediate the electromagnetic interaction between fermions. The QED is a gauge theory,too rather a quantized one.
A: I think that the difference between EM waves and electric or magnetic fields is that we have organs and brain that can directly detect the former (at least for some frequencies). 
For the frequencies that we can not detect, the situation is the same. Our ears can listen to a radio, but we can not sense the radiowave directly. 
It seems that the homing pigeon can sense directly the earth magnetic field. It is surely a "real" stuff for them...
A: Let us begin with a definition, because arguing things without definition may lead to confusion.  An electromagnetic wave is defined as an energy wave that has a velocity in vacuum that coincides with the velocity of light in vacuum. According to this definition, gamma rays, X-rays, ultraviolet rays, visible light rays, infra red rays, radio waves, and gravitational waves are known electromagnetic waves.  Microwaves, alpha rays, and beta rays are not em waves as per this definition.  To produce electric field waves there should be a strain in electric fields, and to produce magnetic field waves there should be a strain in magnetic fields.   If the voltage is increased in a resistance, two successive electrons in serial try to come closer because of voltage, and two electric fields of two electrons should repel each other.  So, there is a strain in electric field.  If the current is increased in a conductor, two successive electrons in parallel try to come closer because of increased current, and two magnetic fields of moving electrons should repel each other.  So, there is a strain in magnetic field.  These are all only situations related with voltage and current in production of electric field waves and magnetic field waves.  In practical situations one need not go for voltage and current.  In a fire, electrons of two successive molecules may bring a strain in electric field and it may produce light (or electric field wave).  When there is a flow of gases in sun, the moving electrons moving in parallel may bring a strain in magnetic field and it may produce radio waves (or magnetic field waves).  So, electromagnetic waves are real.
During one observation, gravitational waves were observed and a gamma ray burst was also observed with a delay of 1.7 seconds.  The magnetic field lines of stars produced gravitational waves just before collision and the gamma ray burst happened after 1.7 seconds during collision.  Microwaves mentioned above are the waves produced by magnetrons; not all microwaves.
