It really existed; in fact, it was part of the appliance package for the higher-end models of all-metal Lustron homes: the Thor Automagic Clothing and Dish Washer!
Even though you changed out the entire wash basket to swap from washing clothing to washing dishes, a lot of people just couldn't feel comfortable about cleaning dungarees and dinner plates in the same machine. It wasn't a success.
Sadly, under new ownership the company tried a comeback early in this century with a slick ventless all-in-the-same-drum washer/dryer and that doesn't appear to have worked, either. The company website is gone and their last official Tweet was in 2010.