What is the ruling on a husband taking his wife back after divorcing her secretly without documentation and after the 'iddah period has passed, and is it possible for the matter to remain a secret between the spouses?

As long as you pronounced the divorce willingly, what you uttered becomes binding upon you. Documentation or public declaration is not a condition for the validity of the divorce. If you took her back during her waiting period (iddah), the return is valid, even if you concealed the divorce and the return from her. However, if her waiting period has ended, you must renew the marriage contract, provided the divorce was not the third (final) one. Otherwise, she is not lawful for you until she marries another husband in a desired marriage.