Finely grate the zest of both limes into a medium bowl.
Cut one of the limes crosswise in half and cut out sections from the fruit from the membranes from one half.
Cut the segments into very small cubes.
Toss the fruit into the bowl and squeeze the juice from the other half of the lime into the bowl.
Add the mango, avocado, onion, ginger, chili, and tuna and stir everything together very gently with a rubber spatula.
Squeeze the juice from the remaining lime into a small bowl and whisk in the olive oil and rum.
Season with Tabasco sauce, salt, and pepper and pour the vinaigrette over the tuna mixture, again stirring very lightly so you don't crush the avocado.
Taste for seasoning; cover the bowl tightly and chill for 1 hour.
At serving time, adjust the salt and pepper and Tabasco, give the ceviche another stir.
Serve immediately, garnished with a few cilantro leaves.