Effects of US Native American tribal membership?

I have been researching my genealogy and have discovered the possibility that one of my distant ancestors may have had some Native blood. Nowadays, the tribe in question has a One Drop policy with regards to membership, so if I can prove any blood at all to the satisfaction of the tribe (right now, not a certainty at all, and I may on the other hand disprove a Native origin as I or one of my relatives researches further). Almost all of my ancestry is White, as far as I can tell.

To give you a perspective, based on the documents I came across, the maximum percentage based on this ancestral line would be no more than 1/64 on my part, which, since the tribe in question has a one drop policy, is apparently enough for me to “be” one of them.

What are the practical effects of gaining and maintaining membership in a Native American tribe that you don’t culturally connect with? Could it affect legal rights or responsibilities in any way?

What legal rights an responsibilities? If you are referring to rights an responsibilities as a tribal member? Ask the tribe. If you are referring to rights an responsibilities as someone before they knew about tribal affiliation? Probably zero.