First computer with real-time clock?

As far as need to keep the time during power off, I don't know that in this case it would matter too much. It would just have to be a combination of hardware and software that kept the current date in the IBM 650. The 650 was small enough that you could power it down during off hours, though I don't know what actual practice was (perhaps it was too much of a chore to boot up so it was kept on all the time). At any rate, as long as the 650 had some way to generate uniform time pulses, a real-time clock/calendar could be implemented, but this would probably have to be at a hardware level since the 650 was primarily a batch machine, or it would have to be programmed to store the date/time in some memory location that the other applications would not touch, and then some counter would have to be implemented that could update a memory location with the number of ticks since the last access or whatever. Of course, as I've been suggesting, there could have been a totally separate clock/calendar device (heck it could have even been mechanical) that kept the local time/date and then transferred it to the 650 upon request. Just like the clock in a modern PC.