power supply OK but no output

I am mystified.  I am installing a simple radio CD player [called "the unit"] as a substitute for the original radio cassette player.  It works [both radio and CD] when I supply 12 volts from a portable power supply, but doesn't when hooked to the cars' 12 volt supply.

Here are the steps I took in installing and troubleshooting.

I tested the unit before taking it to the car.  I supplied 12 volts to both the constant and ignition controlled wiring, hooked up some speakers, turned unit on, got sound out of both speakers.

Then I installed it in my car [Toyota Corolla 1997].  First I checked the voltage on both the constant and ignition supplies and they were both between 12.0 and 12.2 volts when measured to the ground wire.  The "constant" was constant and the "ignition only" supplied power when the key was turned to "accessories" or "engine on".  Everything seemed OK.  I hooked up the power wires, ground wire and front speakers and tested the unit.  The unit turned on, and the unit display indicated the radio was working, but there was no sound.  I tried other speaker outputs and also got no sound.

I gave up, went to bed, had a good sleep, and when I got up the next day did a google search to see if I could find any suggestions.  No luck; any comments related to items that  I had already tested, such as checking all power and ground connections; checking mute, checking fade, etc.

I disconnected both powers and ground from the car supply and provided known good 12 volt supply from a portable power supply.  The unit worked! I tried both radio and CD, and both worked.  All speaker outputs worked. Good sound all around.  Everything looked good.  I returned the unit power and ground wiring hookups to the car supply; and got no sound!  I'm bald, so I couldn't pull anything extra out.

So, I am mystified.  The unit works when powered by an independent source of 12 volts; but doesn't work when hooked up to the cars' power supply.  I don't know what to do next.  If it is pertinent, I have a 2 year diploma in microelectronics and am familiar with troubleshooting in the 0-12 volt DC realm.  I worked in the chemical industry and am familiar with large AC and DC items.  I am not familiar with measuring speaker outputs and only have a simple digital multimeter at home.  Can anyone suggest something?

Thank you, from Harvey in Canada

You are changing two different areas where problems can occur at once; your power supply and your speakers.  Test one at a time;  I would recommend hooking it up to power in your car and testing with one of the known good speakers you used to bench test it.  

I would also suggest you should do a continuity test between your factory speaker leads and your chassis ground, chances are one of the speakers is shorted to ground.  Many times this is the cause of the issue you describe.