Step 1: Make the mojo. Heat the olive oil in a deep saucepan over medium heat. Add the garlic and cumin and cook until fragrant and lightly browned, 2 minutes. Do not let burn or the garlic will become bitter. Stir in the lime and orange juices, water, salt, pepper, and oregano. Bring the sauce to a rolling boil. Correct the seasoning, adding salt or lime juice to taste: the sauce should be highly seasoned. Let cool to room temperature, then stir in the mint.

Step 2: Using kitchen scissors, then a knife, cut each lobster tail almost in half lengthwise through the back (the hard part of the shell). Try not to cut through the soft shell on the bottom. Butterfly each tail, that is fold it open like a book. (The soft belly shell is flexible—to continue the metaphor, it will open like the spine of a book.) Remove the vein (it may be clear or dark) running the length of the lobster tail. If using whole spiny lobsters, cut in half lengthwise and remove the vein. Brush the cut part of the lobster with the olive oil and season with salt and pepper and the lime and orange zest.

Step 3: Set up your grill for direct grilling and preheat to high. Brush and oil the grill grate.

Step 4: Arrange the lobster tails, cut side down, on the grate, running diagonal to the grate. If a crosshatch of grill marks is desired, give each lobster tail a quarter turn after 1-1/2 minutes. Grill the lobsters cut side down until nicely browned, 3 to 4 minutes. Invert the lobster tails and brush with some of the mojo. Continue grilling until the meat is cooked, another 3 to 4 minutes, 6 to 8 minutes in all, basting once or twice more. (Use the poke test to check for doneness—the tail meat should feel firm.)

Step 5: Serve the lobster tails at once with mojo spooned over them and the remainder on the side. Sprinkle the lobster with the remaining mint and serve.