In a small saucepan over very low heat melt the butter. After the butter has melted, skim off all the foam. Add the tangerine juice, whisk and set aside.
Using poultry shears, split the lobster tails in half by cutting the shell down the middle on both sides, and then, with a sharp knife slice the lobster meat in half. Place the lobster tails in a large, resealable plastic bag and pour in 1/2 cup of the tangerine butter. Reserve the remaining tangerine butter. Press the air out of the bag and seal the bag tightly. Turn the bag several times to distribute the tangerine butter, place the bag on a plate, and refrigerate for 10 to 20 minutes, turning the bag once or twice.
Remove the lobster tails from the bag and discard the butter in the bag. Grill over Direct Medium heat, meat side down, for 8 to 10 minutes, turning and basting with some of the remaining tangerine butter once halfway through grilling time. Remove from the grill and serve hot with the remaining tangerine butter on the side.
