Okay I will apologize now for miss understanding some thing here and for having people throw around terms that are incorrect. The signals coming from the stock amplifier are not stereo but are in fact mono, 2 channel mono. Adding an amplifier in line, again a 2 channel amp and inputting a channel to each coil is fine, again providing that inputs and outputs are close to the same. All the way down the line everything is mono. This is easy to check if you have a multimeter and a test tone disk or USB drive. A slight difference not a problem.
The original idea here was to make the factory system better with minimal work. I don't pretend to know everything about the stock amp in this system but like you said in my setup I'm running the exact same setup as the original. I'm just adding the amp in line. Its powering the same signal that the original sub was receiving in the same way.... If the issue of "unique" sound would be an issue with the original system it would be an issue just as much with the upgraded scenario that I now have. At the end of the day this project was about better sound for cheap. I'm being a cheapo here. So if I end up hurting my $70 sub in 3 years bc I listened to a "unique" song, so be it haha. I really don't care. If I was going to spend the money for what these guys were saying to do earlier I might as well have replaced the whole stereo
Where I got confused is when people started throwing around the word stereo. In general you can not apply a left and right signal to a single sub woofer. Bill has a great example above of what will happen. Stereo by it's very definition means different outputs on each channel. An example would be if you spliced into the rear channels (stereo) and ran that wiring to a 2 channel amp and then ran 2 channel out to the speaker, you have problems.
In a case like this you would use something like AudioControl LC2i. It will sum the left and right signal, strip out the high signal and output a bass signal, to a mono amp in this case. Redstapler, I think your amp can do the same thing, you could take left and right signal into the amp and the amp will sum those signals to a sub in the bridged mod.
There are some DVC specialty speakers that are capable of this, primarily speakers for classic cars that only had a single in dash speaker but they have circuitry in them that allows this. This is you may have seen it is okay to route a stereo signal to the speaker.
Again I am sorry we got off on a tangent.