I have the same plant growing here and been trying for a long time to identify it. I believe you are correct in you ID and solve the problem for both of us.
Phragmites australis is listed as a native plant, but recent studies have shown that some stands are an introduce variety and may be the reason for the current invasiveness
Yes, there are some that are native and one that came from Europe long ago that has become invasive: http://www.nps.gov/plants/alie...
I think they are kind of pretty myself, at the bottom of this page there are photo's of the Gulf Coast type and Eurasian type showing the differences in appearance: http://conference.ifas.ufl.edu...
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