How to make a graphical frame when no other graphical frames are present?

One way would be to somehow guess a display and do (make-frame-on-display display). However, (getenv "DISPLAY") does not give a sensible answer (e.g ":" or nil).

I tried to explicitly specify window-system but that also does not work without knowing a display.

One use-case would be to make a togglable persistent emacs frame available on a key press.