Ajitsuke tamago is a traditional dish consisting of soft-boiled eggs that have been soaked in mirin and soy sauce. The egg yolks should be custardy if properly prepared, and the eggs are eaten as a snack, a part of bento, or more commonly, enjoyed as a ramen topping.
The eggs are boiled for a few minutes so that the yolk remains runny, and they are then chilled and soaked (overnight or for a few days) in a combination of water, mirin, and soy sauce. This is the basic preparation, and some cooks might add their own secret ingredients to the marinade to make the eggs more flavorful.