Since the A3 has a software-configurable variable-voltage lighting system for the taillights, and most models are equipped to detect failed bulbs, a solution had to be found that would work reliably, would not interfere with the car's diagnostics, and would avoid problems associated with how the A3 taillights work. I was able to do this using a trailer wiring converter that protects the car's circuitry and puts no load on the lighting circuits. When connected to the correct circuits (not the three tail/brake/rear fog bulbs on each side) it works perfectly, and if any taillight settings are changed (using VAG-COM, for example) it will not affect the trailer interface at all. The converter used is the Hopkins Manufacturing (Hoppy) 46365. To find one, search for Hopkins 46365 or Hoppy 46365.
I have produced step-by-step instructions to guide you through installing the trailer lighting interface on the A3 in a problem-free manner. It's about a 1 to 1.5 hour job. These instructions are also useful for anyone wanting to open up the side panels in the cargo area and/or remove the subwoofer, since they contain details and photos on how to do so.
Click the image below to download the instructions as a PDF. The download file size is about 5MB.
UPDATE - A reader provided the following feedback regarding wire colours on later A3s:
The taillight wire was grey with light green tracer on my 2013. The wire bundle went to a connector where it then switched to grey with brown. Otherwise all the other wire colors were the same.UPDATE 2 - Someone else reported the same wire colour change on a 2011 model, so Audi may have made the change even earlier than that.
Enjoy towing safely with your A3!