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On Cygwin I tried doing ./sage -f mpir and everything broke.
This is because my C compiler uses mpc and mpfr, and on Cygwin (see #25816) at least, it ends up using Sage's mpc and mpfr DLLs. This may be less of a problem on other platforms but I haven't given much thought to it yet.
The mpc and mpfr DLLs in Sage break when Sage's mpir gets uninstalled, so it's necessary at a minimum to also uninstall mpc and mpfr. But at that point it might also make sense to recursively uninstall anything that depends on mpir. This seems reasonable given that they'll have to be re-built anyways.