DISCLAIMER: blah, blah, blah, this page describes modifications I made to the factory wiring on a 2004 RX-8. Not all RX-8s, even of the same year and options package, will look the same as mine, so the instructions below may lead to the destruction of your car (or at least significant damage) if you try to follow them. At the least, you are likely to void all or part of your warranty. Please acquire the factory shop manuals for your specific car to determine how to do this to your car. Remember to disconnect the battery before attempting any of this work and do not bypass the fuses in the car (they are there for a reason, and if they are blowing, it is for a good reason).
This is a scan from the 2004 RX-8 Wiring Diagram. The power for the fog light relay comes from the "FRONT FOG LIGHT RELAY" (circle 150). This is only at +12V when the headlight switch is in the LOW position. The plan is to pull power from another point that has +12V at the times you want it. There are many places you can do this, from hooking directly to the battery so the lights can be turned on at any time (bad idea -- don't bypass the fuse!), to what I did which is to tap into the parking light power (TAIL). Using this later point allows the fog lights to be turned on whenever the parking lights are on, which means also that if you try to leave the fog lights on when you turn the car off, you get the normal warning tone from the car (and hopefully don't kill your battery). I chose to use TAIL instead of the ILLUMI, because the parking lights draw much less power than the interior illumination does and I didn't want to overload a circuit.
This drawing from the 2004 RX-8 Wiring Diagram book shows the wiring for the fog light relay, switch and lamps. The power to actually drive the fog lights goes through the FOG fuse. What we are doing is to provide a new power source for the control side of the relay. When the fog light switch is turned on, it completes the circuit which closes the FRONT FOG LIGHT RELAY which completes the circuit to actually power the fog lights.
The wiring work will be done inside the fuse box in the front driver side of the engine compartment. We are going to be modifying the wiring to the FOG relay (on the right center) by pulling power from after the TAIL fuse (little fuses top center).
This is the inside of the fuse box. The FOG relay is the lower of the three black boxes on the right side of the fuse box. The TAIL fuse is one of the colored guys top center. To remove the fuse/relay holder (tan plastic), there are a few snap connectors around the perimeter. You should not need to remove any wires or nuts/screws.
The brown wire is the hot side of the control side for the fog light relay. I cut this and taped off the end in an appropriate manner (remember that it will be at +12V whenever the low beams are on, so you don't want it flopping around in the fuse box). I spliced the tail of the brown wire to the black-with-red-stripe wire from the TAIL fuse using a cable splice.
This is a shot with the parking lights and fog lights on.
This is a shot with the fog lights and high beams on.