I tried to do

      - id: cygwin
        uses: cygwin/cygwin-install-action@master
        with:
          packages: ...
      - name: build
        run: ...
        shell: ${{ steps.cygwin.outputs.root }}\bin\bash.exe -leo pipefail -o igncr '{0}'

but apparently hit actions/runner#444.

What I propose (and can make a PR for if you like) is what setup-msys2 action does: create a directory under ${{ runner.temp }} with a .cmd script that invokes the correct bash, and adds that to the PATH, so that subsequent steps can use shell: cygwin.cmd '{0}'.