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It would be a miracle if we could build working lasers and computers on the assumption that electrons exist if electrons didn't really exist.
Any observational data can be explained by any number of different, mutually incompatible theories. So why think yours, that posits unobservable entities, is the right one?
Realists believe that the aim of science is to offer a true theory of all aspects of reality, what is observable as well as what is unobservable.
Anti-Realists believe that the aim of science is to offer a true theory of the observable world, the unobservable world being beyond our ken.
Not all mutually incompatible theories are equally good. Some are better for being simpler or explaining the data in a more intuitive way. Some posit fewer hidden causes or are just more plausible.
So it isn't really true that there are always equally good alternatives.
Why should simpler or more plausible theories be more likely to be true?
Historically, there have been many empirically successful theories that later turned out to be false. Phlogiston and the Wave Theory of Light are examples.
If the Anti-Realist is right about the underdetermination of theory by data, it applies to theories about observational data as well as to theories about the unobservable, but the Anti-Realist is happy with theories about observables.
The underdetermination argument may just be a sophisticated version of the problem of induction.