Electrical Issue: Car radio losing presets and factory clock resets when I turn vehicle off sometimes. Why?

There are two power sources to the factory clock and the radio, one is the "always on" which supplies power to keep the radio presets in memory and the factory clock accurate, and the "key on" which powers the factory clock display and full operation of the stereo.

Just recently, the "always on" power became inconsistent, usually supplying power, but sometimes interrupts when I turn the vehicle off. It can't be the fuse, because if it were, power would remain off. I'm thinking it's a relay, but I can't identify which relay it is from both fuse boxes. Everything functions perfectly when vehicle is running (except having to reset clock and radio presets again).

It's a 1996 Honda Accord I've owned since 2001, without any previous electrical problems. I just installed a new battery (factory from the dealership) a month ago, and installed the aftermarket radio six months ago (I installed both myself, and I made certain to install the radio exactly according to the directions). But I've only had this problem for the last few days.

If it's a relay, what would it be called? Or could it be something else?

Need a new (recharge) car battery, perhaps.