Cook the garlic in a large saucepan of boiling water for 3 minutes to blanch. Transfer the garlic with a slotted spoon to a food processor or blender, then set aside until needed. Add the beans to the boiling water and cook for 4 minutes or until tender but still a little firm. Remove the beans with a slotted spoon, rinse under cold water, then drain. Add the potatoes to the saucepan and cook for 12 minutes or until tender, then drain.
Add the vinegar, mayonnaise, oil and a pinch of salt to the garlic in the food processor, then blend to a purée. Add the basil and water, then blend again.
Transfer the garlic-mayonnaise dressing to a large bowl. Add the tomatoes, beans, potatoes and tuna, tossing to coat. Season with pepper, then top with the olives just before serving.