"I see no thermostat? What turns it on & off?"
Sorry, I wasn't clear. That is the switch T_Bone and I added, in the photo on the right of the heat pump. My lovely wife hasn't complained all these years of not having automatic hot water.
It was only when I replaced a leaky tank heater that I discovered the mismatch between components (heat pump and new tank heater). With a new tank heater just installed I preferred not to remove it and buy another one with a different control.
Which probably is why manufacturers now mostly (only?) only sell tank heat pump water heaters, not add-ons. I wouldn't mind IF the tanks lasted like the ones of old. I got 13 years out of the heat pump, haven't had a tank last that long recently.