As mentioned above, I had a similar "fly line" experience recently in NAS. Two of us traveling together but in different security lines were both directed to either strip the lines off the reels we had in our carry-on bags or check the entire carry-on bags as checked baggage. We both elected to strip the fly lines off--leaving them at the gate--and carried the reels--now with backing only--inside our carry-on bags without further incident.
I've traveled to the islands every year since 911 and only during the first post-911 flight did I ship all my tackle as checked baggage. On all other trips I've carried enough gear on board to fish, including at least one and sometimes two 5-piece rods, one or two reels (with fly lines), leaders, tippet, and flats clothing and wading shoes--all with no problems. I also carry about two dozen flies in a zip lock and show them as I pass through the TSA gate. All my other gear including 4-piece rods, back-up reels, flies, tools, spare lines etc. goes in my checked baggage.
Like many others posting here, I don't have a problem with the TSA procedures and the need for them to protect us all. I do sometimes lament the inconsistency of their enforcement, but, in general, I try to work within the guidelines. I know others have had the reel and fly line issue come up during outbound flights at Nassau and I just heard that Ted Lund had the same problem from outbound security leaving Guatemala on a recent trip. I have not, as yet, heard of any fly line restriction problems occurring for travelers leaving the US. So for now, I'll likely start stripping my lines off on return trips and see if that satisfies the issue.
For those who suggest shipping all fly gear as checked baggage, I understand their choice. But I am not willing to take the risk of getting to my destination without enough gear to fish. So I take a different risk, knowing I may encounter carry on problems and have to check additional gear at the gate or even risk losing an item.
I guess the best thing all of us who travel by air to our favorite fishing spots can do is post our incidents on forums like this one, so we can all stay as up to date as possible on the changing security regulations and the changing interpretations of the rules at different destinations. That way, hopefully, we can each pack our stuff to meet whatever risk profile we choose to undertake.
Dick