Cover eggs with cold water by 1 inch in a 2 to 3-quart saucepan and bring to a rolling boil, partially covered. Remove from heat and let eggs stand, covered, 10 minutes. Transfer eggs with a slotted spoon to a bowl of ice and cold water and let stand 5 minutes (to cool). Gently tap shell all over with back of a spoon to lightly crack (do not peel). Do not tap too hard or tea liquid will seep into shell instead of just staining cracks.
Bring soy sauce, sugar, and water to a boil in saucepan, stirring until sugar is dissolved, then add tea bags. Reduce heat and simmer, covered, 10 minutes. Add eggs (and more water if eggs are not completely covered by liquid) and simmer, covered, 10 minutes. Remove pan from heat and let eggs stand in liquid, uncovered, until cool, then chill at least 2 hours. Lift eggs from liquid and peel. Reserve 2 tablespoons cooking liquid and discard remainder.
Whisk vinegar and reserved cooking liquid into mayonnaise and serve with eggs. Present eggs whole, then quarter for dipping. (Yolks may have a dark ring.)