Atozuke
Atozuke 「後付け」 is the state of a yakuless tenpai hand, that gains yaku upon the claim of a winning tile. In other words, the hand currently does not possess yaku, but it is tenpai. However, this term does not necessarily apply to closed hands, whose only yaku is mentsumo. In the case of mentsumo, any tile produces yaku upon tsumo, which is not what atozuke implies. The use of atozuke is subject to a variable rule, which may or may not allow its use. That decision falls on a league, organization, or house rule.
Cases
Atozuke may be employed in a number of hands.
Agari:
or
This hand uses a shanpon tenpai for two tiles. One may produce a yaku, while the other does not. In the event of haitei, houtei, or even rinshan, the issue of atozuke here would become moot, as yaku may apply to either winning tile.
Risk of furiten
Draw:
Particularly, open yakuless hands run the risk of furiten. That is especially true if the above example draws a 9-pin during the course of the hand, when additional tile draws remain in the wall. In this case, a player cannot declare a win and must discard any tile in the hand. Upon doing so, the player become furiten and must work around it, by changing its hand composion with the remain tile draws or other player discards.
Sakizuke
Sakizuke 「先付け」 is the complete opposite to atozuke. In this case, the practice of using atozuke is disallowed. With this rule in place, a player would have to take greater caution making open calls by either first ensuring yaku or handling the case of furiten.