Based on answers from:
Do electric and magnetic lines of force physically exist?
Is the concept of a field necessary to electrodynamics?
Why isn't the electric field just a mathematical tool?
It is real. Whatever "real" means. The ones I liked the most:
The reason one might expect such a description to be permissible is that there are a lot of formal similarities with the equations of Newtonian gravity, which is an action-at-a-distance theory where it's possible to dispense with the inter-particle force concept and instead use a field theory. This is permissible even though the gravitational field doesn't physically exist, because the two procedures necessarily yield the same values when you try to compute an object's acceleration. When they realized E&M fields store and transmit energy and momentum, people quickly began to accept them as physical realities. This sparked a lot of interest in looking at Newton's law of universal gravity in a new light. The goal was to either to modify it to bring it more in line with the format of Maxwell's laws, or at least isolate any fundamental reason explaining why you can't. A young German physicist came up with a solution to this problem and an improved theory of gravitation in 1915.
The electric and magnetic fields are real things: they can store energy and transfer momentum. And, yes, the electromagnetic interaction can be described in another (more fundamental) way as exchange of bosons in a quantum field theory. But that does not change the fact they these fields store energy and transfer momentum.