I'm guessing its an electrical design issue with Japanese cars.
your hid fogs are after market, and i assume the ballasts need a quite high voltage to start the "High Intensity Discharge".
you see the low beam hids already need vast amount of the design voltage to start while ignition (hence the low voltage look to all the lights at one instance), the battery just can't take that much. and when the alternator kicks in, everything's gone normal (hence turn the fogs off then on it works).
It's the same issue with CD players, with Japanese cars when you start the engine, CD pauses/shuts for an instance to allow all the voltage to go igniting the engine and lights and stuff.
I have been driving bmws and mercs, the CD functions normally when turning the key.
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