Yes, "real" is a tricky word. It's a word from the domain of the philosophers. There is a philosophy of science called scientific realism which claims that the entities we talk about in science are "real." This is contrasted with scientific insturmentalism which argues that these entities are merely useful instruments for describing the world. Most of the time we don't worry about such philosophical demarcations. Most of the time the value of science is its ability to explain the world around us and to make predictions about what could happen in the future (such as in future experiments).
As such, the best approach we can take to these questions is to look at several related phrasings. We can say rather confidently that the electric field is at least as real as the gravitational field (I say "at least" because general relativity makes really interesting arguments about it). But what about forces? Is the force of electrostatic repulsion "real?" Are any forces real? Truthfully, the concept of "force" only came about through Newton and was only possible to pen down precisely thanks to the invention of calculus. Before that, any use of the word "force" was your garden variety natural language sense with all its gloriously fuzzy meanings.
But let's give Newton the benefit of the doubt. Let's say his concept of "forces" is "real," and electrostatic repulsion forces are real. When we construct the equations for electrostatic forces, we find that they are conservative. This means that if you take any particle around a closed path, no matter how complex, and look at how much work is done to the particle, it is 0. For sake of argument, take an electron in the presence of a highly negatively charged object. As you move the electron towards the plate, it takes work. As you move the electron away from the plate, it does work on you, and they always cancel out, no matter the path. That's a conservative force.
If you have a conservative force, you can do some calculus to show that there is a scalar field associated with it: the electric field. This is a real number associated with every point in space. To determine how much work is needed to move a particle along any path (closed loops or any open path), you can just look at the value at the end points, multiply it by the particle's charge, and subtract one from the other (I'm handwaving a minus sign for simplicity), and you come up with how much energy it will take to move the particle. Obviously this is consistent with the definition of a conservative force: if you look at any closed path, it starts and ends at the same point, so the work required will always be in the form $x-x$ which equals $0$.
Is this real? That's a philosophical question. The predictions it makes are certainly as real as the underlying assumptions (of conservative forces). But we can note a few interesting things. The first is that this electric field describes the real forces that will be applied to any charged particle (remember, we're assuming forces are real here). That's pretty real. It also describes the forces that would be applied to any hypothetical charge that might exist. That feels less real, more predictive, but you can be the judge.
There are some particularly interesting aspects of how it interacts with magnetic fields. If you work out Maxwell's equations, you find that it's possible to have a constantly varying electric field and a constantly varying magnetic field that pass energy back and forth as they travel at a high speed. In fact, that pairing travels through space at the speed of light. That's Maxwell's description of what light is: just a little slug of energy bound up between those two fields, traveling onward. Is that "real?" Or is that a "photon?"
It turns out that that's where quantum mechanics starts to play into things. Before QM, there were experiments which demonstrated that light behaved like a wave -- the way Maxwell predicted. There were other experiments that demonstrated that light behaved like a particle -- a photon. This dichotomy persisted until quantum mechanics showed that light was better thought of as a quantum mechanical entity that behaved somewhat like a wave and somewhat like a particle, and we just had to deal with that.
Now one of the most successful quantum mechanical theories, the "Standard Model" will argue that there is a "photon," the quantum mechanical version of it, and that photons are the "carriers" of the electric force. It models all electrostatic interactions via the emission and receipt of these photons. Quantum Field Theory, which unifies the standard model with special relativity treats said photon as just an excitement in yet another fundamental field. I write this because you asked about how QM treats electric fields, but I don't think very much insight can be gleaned from these quantum mechanical treatments unless one has the substantial body of mathematics and theory needed to leverage them.
So in the end, that's a lot of words to not answer your question of "is it real?" But hopefully somewhere along the way you come to your own conclusion about the answer you wish to reach.
