I have the same problem and have been doing some research. It seems that some of the nerves that control the sphincter between the esophagus and stomach (LES) could be compressed through TOS and cause the LES not to function properly. It doesn't seem like much research specifically correlating the 2 has been done but it's possible.

I have been suffering with GERD for over a year. I've changed my diet, I've taken high doses of prilosec and zantac, I'm not overweight... Nothing helps - so maybe it is my nerves. Anyway, thought I'd post this in case anyone else is dealing with it.