Love my "new" 91 Honda Civic Hatchback....however:
When I turn on my head lights, my tail and dash lights go out.
When I turn my head lights off and only have on my front corner lights, my dash and tail lights come on.
When I turn off my head lights and only flip my high beams, the dash and tail lights come on.
Head lights on + High beams=no dash and tail.

What is this? Please someone say "don't worry, it's easily and cheaply fixable."

 

Had the same problem in my 89 h/b. In my case (same problem) it was a loose fuse in the #8 spot (10 amp) under the dash. Pull the fuse and slightly twist the prongs which will provide a better, tighter contact when replaced. After I did this, have had no problems since. Good Luck.

 

Forgot I have been cursed with a Honda Motor Company of Canada civic so the #8 fuse may not be in the same location, but it will be labeled as tailight/ instrument panel.

 

Sound like it my be a relay problem you got there.