In principle, we could describe all physics without EM fields (or photons), as they are mainly a useful tool to describe "action at distance" (which does not mean instantaneous) between charged particle. In some sense, I could always integrate out photons, and describe only electrons and not have any changes in the observations (as detecting photons is done by observing how the motion of charged particles changes). With this picture, all photons are "virtual" (in the QFT sense), and we could expect them to always be off-shell.
My question is : why do we expect that photons that are "really emitted" (in a sense that should be made clearer, but that might mean "travelling very far before being absorbed") are always on-shell (i.e. having $E=p$)? Is it because the propagator of an off-shell photon decays very fast and therefore these photons can not interact with long distance charges ?
We could imagine that two very distant electrons (say, in two different galaxies) "scatter" each other (what we usually call "seeing a distant star") with off-shell photons. Why is it not so ? Is it just because the probability of this event is very small ?