Looks like you have done lots of your own homework: good job!
If you are receiving income from the work you are doing abroad for this business, you are generally going to get taxed on that income in the place where you work. How do you get paid from your LLC? Management fees? This might reduce your tax liability and make life easier.
On the other hand, does the LLC make a profit and distribute profits to you? This might change things. I'm guessing the LLC is a Florida LLC?
You'd probably do best to have consultations with a couple lawyers to get a better grasp of your specific details. In any case, your LLC will likely need to file for taxes.
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If is a multi-member LLC as you mentioned, it is not treated as a disregarded entity, like a single member LLC would which means that instead of income reporting on a single member's personal taxes, the LLC itself would need to pay taxes. Does you LLC have a tax id #? As Counsel mentioned, the best thing to do is speak to a Tax/Business Attorney locally. Good luck to you.
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