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There is this fabric looking thing protruding from between the A-pillar and weather stripping. It is above the side view mirrors, on both sides, and has always been there. What is it? Is it typical (2006 GT)?
Your mirrors gasket is what you are seeing. You can pop the backing covers off, and you will see 3 nuts (11mm nuts) that you can loosen and set your mirrors so that they are aligned better. Yours looks out of alignment, almost loose. Hope that helps you.
it looks like it may be some of the foam from under the pillar cover that got stuck under the weatherstrip. You may have had it from day one and never noticed.
you may have a better idea if you pop off the a pillar cover to have a looksie.
I have the same black felt cloth looking thing sticking out by my driver side window on my 05. I am assuming it's some sort of insulation or something.
There is this fabric looking thing protruding from between the A-pillar and weather stripping. It is above the side view mirrors, on both sides, and has always been there. What is it? Is it typical (2006 GT)?
Yes, that is typical. My 07 had it too, about in the same spot.
It's just some insulation stripping. You can fix it by removing that trim, which comes off from inside the car.
IIRC, you'll have to remove your interior a-pillar cover to get to it.