Dice the onions, pepper and apple as finely as you can.
Open the cans of tuna, add them to a bowl and use a fork to flake them apart. Add the mayonnaise and mustard, stir, and then mix in the pepper, onion and apple.
Add the salt, pepper and juice from the entire lime. Taste for seasoning and adjust according to your liking.
Set the tuna salad aside and prepare to make the grits. It takes a lot of time and attention to make sure they're smooth and creamy, so I don't suggest making them at the same time as the tuna.
Bring 3 of the 4 cups of water to a boil, then add the salt and grits, pouring slowly to discourage lumps. Stir regularly until the grits begin to thicken, then slowly add in the remaining cup of water, about a quarter cup at a time. This is another technique of mine to discourage lumps, although you could try boiling all 4 cups at once.
Keep regularly stirring the grits until they've completely absorbed the water. The whole process will take about 20 minutes. Stir in the butter, and serve alongside the tuna.