Does M-: (python-shell-completion-native-try) RET in the *Python* buffer
return nil?

Perhaps try putting some `message' calls into
python-shell-completion-native-get-completions? (Using the debugger is a
bit tricky due to the timing of process communication.)  Especially
check the contents of the current inside the `when' block at the end:

              ;; Grab output until our dummy completion used as
              ;; output end marker is found.
              (when (python-shell-accept-process-output
                     process python-shell-completion-native-output-timeout
                     comint-redirect-finished-regexp)
                (re-search-backward "0__dummy_completion__" nil t)
                (cl-remove-duplicates
                 (split-string
                  (buffer-substring-no-properties
                   (line-beginning-position) (point-min))
                  "[ \f\t\n\r\v()]+" t)
                 :test #'string=))

Try the recipe in https://debbugs.gnu.org/cgi/bugreport.cgi?bug=28051#8
and see if you suffer from that bug.