In the “real world,” what tasks are Mitty and his wife carrying out?
The tasks that Walter Mitty and his wife carry out in the real world are presented as being as mundane as possible. This is a deliberate strategy on Thurber's part to provide a stark contrast with the contents of Mitty's exciting daydreams. Indeed, Mitty only retreats into his dream world because he finds real life so incredibly boring.
Heading into the city for a shopping expedition is an incredibly boring prospect for Walter. It's fair to say that Walter doesn't go in for retail therapy all that much, so the thought of accompanying his wife while she visits numerous stores doesn't exactly float his boat, so to speak. Truth be told, it positively invites him to retreat into this dreamworld.
His first such foray into fantasy comes when Mitty and his wife are driving into the city. The dullness of this drive is contrasted with the excitement of Walter's daydream, in which he's the commander of a military aircraft, standing brave and resolute while his crew are scared stiff.
Later on, right at the end of the story, Mitty's wife pops into the drugstore for one last item before they had back home. Not surprisingly, dreamy Walter's over-active imagination conjures up yet another fantasy. This time he's in front of a firing squad, smoking his last cigarette before being executed as a traitor.