Of course they're real - as real as (say) velocity is. Like velocity, its measurement is dependent on the observer frame; only here, "frame" doesn't just mean a speed reference or a location in space to use as an origin for a coordinate grid or a reference time to use as time 0, but also a gauge; specifically, the ground for potential. Just because something is relative doesn't mean it doesn't exist. Otherwise, you'd be saying that velocities don't exist because they're relative ... or that phase doesn't exist, because it's also relative ... when (in fact) phase is practically everything as you can see in this demo here.
A Scalographic Demo
The phase (color coded) is, itself, what determines the scalogram.
https://www.youtube.com/watch?v=OugT7uGGtNg
[Also: in quantum representations, in case anyone forgot, the electromagnetic potentials show up as coefficients for the expression of the phase.]
Contrary to what you may see in some folklore, sometimes even propagated by physicists themselves (who should know better!), both the magnetic potential $𝐀$ and the electric potential $φ$ have always been present in Maxwell's formulation of electromagnetic theory from the very first day - meaning: even the papers he wrote, predating the treatise on electromagnetic theory (e.g. his 1864 "A Dynamic Theory of the Electromagnetic Field"). They were not put there as afterthoughts by later people. Maxwell called $𝐀$ the "Electromagnetic Momentum" and was accurate in that call, because the canonical form of the force and power law for an electric charge $e$ with momentum $𝐩$ and energy $H$ can be written as
$$\frac{d}{dt}(𝐩 + e𝐀) = -∇U, \hspace 1em \frac{d}{dt}(H + eφ) = \frac{∂U}{∂t}, \hspace 1em U(t,𝐫,𝐯) = e(φ(t,𝐫) - 𝐯·𝐀(t,𝐫)).$$
So, it plays the same role with respect to momentum, that the electric potential plays with respect to energy - it is a "potential momentum" per unit charge.
It was only in the later part of the 19th century, during the Wars of the Diadochi period, following Maxwell's early demise, as his successors carved up his realm between one another, that you saw people (meaning: Hertz) trying to push the potentials away as if they were some kind of fiction and trying to write everything in terms of the field strengths alone. It is partly due to the misguided efforts of these later 19th century successors that the notion entered the folklore that potentials somehow weren't real.
A clear hall-mark of that being the wrong approach is that you can't write down any action principle. Everything's in the wrong places. You'd have field strengths as fundamental variables, so then what order would they appear as in, in the Lagrangian?
Dirac and Bohm, in the 20th century, put a stop to the "potentials are fiction" meme with their respective analyses (to say nothing of Weil).
Even today, Physicists have not fully learned that lesson or to pay attention to clues like that, because the same thing is still happening - with the Dirac equation and its action principle. There, too, you find an odd-ball Lagrangian that is totally zeroed out on-shell - which is a dead giveaway that you have order 0 and order 1 terms mixed up together in the wrong place.
Unbeknownst apparently to them all, this too can be written in terms of "fermionic" potentials, with the spinor field being the field strengths of the potentials. When written this way, the Lagrangian becomes quadratic and can even be factored as a perfect square, similar to what happens with the Maxwell-Lorentz or Yang-Mills Lagrangians. And here, too, there is a gauge condition for the potentials; and there could be some interesting physics here that has yet to be explored.
So, one should also be talking about fermionic potentials, as well; not just gauge and electromagnetic potentials; and, if anything, people have (even to the present day) yet to realize the full potential of potentials.
