Prior to the fall, rule is not mentioned at all, except that both are called to have dominion over the earth. And after the fall. Gen. 3:16 is the only verse to mention rule, and say it will happen.
Gen 3:16 Unto the woman he said, I will greatly multiply thy sorrow and thy conception; in sorrow thou shalt bring forth children; and thy desire shall be to thy husband, and he shall rule over thee.
Note that God did not say he should (or is supposed to) rule, only that he would (is going to). Would is a prediction, should is a command.
Reading that prediction as a command is like reading that Satan (the snake) should (is supposed to) bite the heel of the couple’s children (Gen. 3:15), and to read that God would be pleased with Satan for doing so.