What our successful scientific theories say about unobservable entities and processes is approximately true.
This is not the only way to formulate scientific realism, but it is one of the more-often used ones, and it does effectively separate realism from versions of anti-realism which hold that we are not justified in believing what our theories say about unboservables.
Obviously, this version of Scientific Realism cannot be directly tested using our current theories and current technology, since what is currently unobservable can't be observed now.
However, what is observable shifts over time (at least in one important sense of the word 'observable'). This can happen either because (1) we develop the ability to reach new regimes of old variables (e.g. scientists create technology to make materials colder or hotter than we previously could, or we can study bodies moving at higher and higher velocities), or because (2) scientists develop new instruments that enable new types of observation reports (e.g. telescopes, microscopes, fMRI machines, or mass spectrometers).
This suggests a way to test realism diachronically, using the historical record. First, find something that went from being unobservable to being observable. Then find theories that were (considered) genuinely successful at that earlier time, and see what claims it made about the previously-unobservable-but-now-observable world. Finally, check those claims against the now-observable reality.
Scientific Realism (at least the version stated above) predicts that the old claims about the previously-unobservable things will usually approximately match the new observations of those things. (I say 'usually' instead of 'always,' because sensible realists are fallibilists.)
I have not run this test myself. To do it in an intellectually responsible way, a large survey of past transitions from unobservable-to-observable would have to be collected, and step would have to be taken to make that sample of transitions representative. However, at first glance, it looks like at least some cherry-picked famous examples don't bode well for the realist's prediction:
- The telescope played a significant role in the scientific revolution
- The vacuum pump played an significant role in the scientific revolution
- The ability to cool things down further and further led to the discovery of superconductivity
- The ability to study bodies at higher and higher speeds was crucial in the transition from classical mechanics to special relativity
There are historical examples that run in the realist's favor too; I think one good example is that (on the whole, i.e. usually) phylogenetic trees generated via molecular data matched previously existing phylogenetic trees fairly closely (i.e. the old trees were usually 'approximately true,' which is all the realist wants). This is why, as I said, we need a large survey to figure out which historical transitions reflect the overall, general pattern, and which cases are outliers.
Finally, in terms of already-existing arguments, this is not really very different from the Pessimistic Induction (if at all). I think of it as a specialized version of that argument, focusing on the realist's claim that the observable/ unobservable boundary does not mark an epistemically important distinction. For this reason, I think of the above as a diachronic version of Kitcher's "Real Realism" (which potentially comes to the opposite conclusion of Kitcher's view).