Nevertheless falsification is surely the most coherent justification for scientific reasoning in the sense of "a philosophy professor has a gun to your head and demands that you justify your work as a method for approximating reality"?
No, if anything falsificationism and scientific realism are in tension. From Popper's The Logic of Scientific Discovery:
> I think that we shall have to get accustomed to the idea that we must
not look upon science as a ‘body of knowledge’, but rather as a system
of hypotheses; that is to say, as a system of guesses or anticipations
which in principle cannot be justified, but with which we work as long
as they stand up to tests, and of which we are never justified in saying
that we know that they are ‘true’ or ‘more or less certain’ or even
‘probable’.
The most common realist arguments are probably variations on the "miracle argument": our scientific practices work, and the only plausible explanation is that they have at least some ability to find the truth.
Isn’t that a refutation of justificationism, not of realism? Isn’t he just saying that there’s no process you can follow that will guarantee that you haven’t made a mistake, or similarly, that there are no claims which can reach the status of being beyond criticism?
This can be said of all of philosophy (at least outside of the purely formal world of logic), not just the physical sciences. From that perspective, 'standing up to tests' is a poor way to evaluate hypotheses, except in comparison to all the others.
Paying attention to falsification also performs the useful task of pushing hypotheses into saying something definitive.
Science is much more than falsification, of course: for one thing, a falsificationist stance does not generate hypotheses on its own.
I'm not suggesting that all of philosophy should adopt falsification as a principle (and much less that it should be ignored if it cannot): ethics, for example, is a field which I think is important beyond what is falsifiable.