We honeymooned in Anguilla last September - a risky time of year with hurricanes, but we trusted that our time would not be disturbed - and it was not.
We rented Chinaberry, a beautiful villa that allowed us easy access to Rendezvous Bay, and we would walk the beach every day. We never encountered more than 10 people on the entire length of the 3-mile beach, and typically found our footprints to be the only tracks to be found (which was very special, as we had received flip-flops that left imprints reading "Just Married" as wedding presents).
Most restaurants were closed, but those that were open were filled with friendly locals instead of tourists. For those who desire even more seclusion than usually found on Anguilla, September can be great. For a honeymoon, the whole island is perfect!
