cache trying to use git for windows' tar.exe on self-hosted runner, failing to find correct gzip

On a self-hosted Windows runner, I just noticed that caches are failing now with errors like:

"C:\Program Files\Git\usr\bin\tar.exe" --posix -cf cache.tgz --exclude cache.tgz -P -C C:/runner/_work/msys2-autobuild/msys2-autobuild --files-from manifest.txt --force-local -z
/bin/sh: line 1: gzip: command not found
/usr/bin/tar: cache.tgz: 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

Caches used to work fine, using the version of tar.exe in system32. My understanding is that the programs in git for windows usr\bin are supposed to be regarded as internal to Git for Windows and should not be relied upon by external things like this. @dscho?

Current theory is that Git for Windows' tar is looking on the path for gzip, either not finding it in my case, or in other cases finding one incompatible with that tar (due to being based on a different Cygwin).

No file or test is named in the report. Begin by reproducing the shown cache-save command on a self-hosted Windows runner, then trace how the toolkit selects tar and resolves gzip. Done means cache saving no longer fails with the reported missing-gzip and exit-code-2 errors.