I have the same problem in my 95 Ls400 with the Radio LCD, the tuner display is dimly lit (buttons not backlit) when the headlight are off. When I turn the lights on, the radio light goes out. The buttons are never lit. The radio spontaniously lit up one night while I was driving, with all the buttons back lit, but has never done so again. Seems like some sort of electrical short, but I'm not sure what to look for. My climate controls are lit up properly.
Thanks!