GitHub's @actions/cache library, used by the actions/cache action, is hardcoded to use Git for Windows' tar. That's a problem if you install Cygwin and put Cygwin's binaries in the PATH, because the Git for Windows includes older versions of the Cygwin libraries, GfW tar expects to be linked against those, but actually finds the freshly installed Cygwin DLLs, and promptly falls over.

Run actions/cache/save@v3
"C:\Program Files\Git\usr\bin\tar.exe" --posix -cf cache.tzst --exclude cache.tzst -P -C D:/a/cygwin-install-action/cygwin-install-action --files-from manifest.txt --force-local --use-compress-program "zstd -T0"
/usr/bin/tar: cache.tzst: Cannot write: Broken pipe
/usr/bin/tar: Child returned status 127
/usr/bin/tar: Error is not recoverable: exiting now
Warning: Failed to save: "C:\Program failed with error: The process 'C:\Program Files\Git\usr\bin\tar.exe' failed with exit code 2
Warning: Cache save failed.

This repository hopefully provides a simple test case to demonstrate the issue.