Which brings up a, not so hypothetical, question. If an MM from Arkansas visits a lodge in a blue state, what happens?
Nothing hypothetical about that. Visits of that sort happen all of the time.
As Arkansas does not recognize PHA in Arkansas they don't recognize PHA anywhere. This means the lodge he is visiting is a "George Washington Affiliation" lodge for like of an official term (I refuse to use the word mainstream as both branches are mainstream and always have been).
If no PHA brothers are visiting, no issue.
If yes PHA brothers are visiting then it depends on the Arkansas rules. I know the rules in my three jurisdiction therefore I expect Arkansas members to know there own rules. There are "act like the locals" rules and there are "act like you are at home" rules. If the Arkansas rule is to act like the locals, stay in the meeting. If the Arkansas rule is to act like you are home, leave the meeting (and then go through the line in an Arkansas line so you can submit legislation to change the rule to prevent future repeats of this embarrassment).
By the way this happened at the GL level in Alabama several years ago. The GM of Alabama visited the GL meeting in Arizona. Of course the PHA GM of Arizona also visited as cross attendance is standard in states with recognition. The GM of Alabama had to leave. The Internet went wild laughing at Alabama that year and their rules roundly desired that laughter. So they voted in a change to "act like the locals". Two years later I was at an Illinois GL meeting. Of course the entire PHA grand line was in attendance as usual. The GM of Alabama was the invited keynote speaker. He visited and spoke no problem as his rules had been recently changed. Some of us who applauded him applauded knowing their rules change. Some at the meeting just applauded because every Mason applauds every GM for everything.
It's all simple and a matter of looking up your rules. And of getting active changing hose rules if they are not what you think is right.