“Live local.” Is it a complete sentence?

I guess it's like "Think different." I don't have an answer but I would be curious to know if "Live local" would be grammatically correct.

Yes, it's a complete sentence. It's an imperative, urging you to, perhaps, participate in matters relevant to your local community and/or stimulate your local economy. I don't know your context, but I imagine it's something like "Don't travel to the Riviera; go to your local beach. Don't buy cheap foreign knock-offs. Forget the Louvre; go to a nearby museum. Eat at your local restaurants. Join the PTA. Live local."

It's worth mentioning that in "live local" (like in "think different"), "local" is an adjective that's being used as an adverb, ie "live locally", "think differently."