I believe my real analysis class made me a significantly better programmer and thinker in general, but I don't use anything specific from it (not yet, anyway).
Genuinely curious how real analysis made you a better programmer...? I've taken it and loved it, but my interest was more due to physics/engineering applications. I can't see a connection to programming.
I suspect it could be that real analysis truly makes you challenge your assumptions and forces you to build a habit of thinking through corner cases.
Other maths courses do this as well, but real analysis is particularly vivid for some people because of all the fun counter-examples you get to see in a well-taught course, such as a function that is everywhere continuous but nowhere differentiable.
This habit of thinking through corner cases is something I miss from a lot of (junior) programmers.