This achievement can only be obtained after getting the Divine Mirror (which is obtained in Chapter 8).
Each tail is acquired by defeating specific officers on the Mirror Realm version of stages in Chapter 5, 6, and 8. Press the Y button on the stage select screen to switch between Normal and Mirror Realm versions, just like you would to get to a Redux mission.
Chapter 5 - Struggle at Nanjun (Mirror Realm) - NOBUNAGA ODA must be defeated and shows up with DA JI after CAO PI is defeated. This one is unmissable since he must be defeated to achieve the victory condition.
Chapter 6 - Defense of Tetorigawa (Mirror Realm) - NAGAMASA AZAI must be defeated and shows up at the very end of the mission in the enemy's main camp. This one is also unmissable.
Chapter 6 - Rescue at Nan Province (Mirror Realm) - ZHANG JIAO must be defeated at the very end of the mission. IMPORTANT: YOU MUST NOT DEFEAT MUSASHI MIYAMOTO when he shows up again at the END of the mission (He will join you after finding out about ZHANG JIAO). However, you should defeat MUSASHI MIYAMOTO at the START of the mission. This one is MISSABLE if MUSASHI MIYAMOTO is defeated before ZHANG JIAO when they both show up at the end.
Chapter 6 - Investigation at Mt. Niutou (Mirror Realm) - PANG TONG must be defeated at the end. This one is unmissable.
Chapter 6 - Intrigue at Xuchang (Mirror Realm) - This one has TWO TAILS in it. SIMA ZHAO and YAN LIANG must be defeated. SIMA ZHAO is very close to where you start this mission and should be taken out right away. YAN LIANG is easily missable and starts in the western garrison with GOEMON ISHIKAWA. Simply head over to the western garrison after defeating SIMA ZHAO and take him out for the second tail. This mission has one missable tail (YAN LIANG). SIMA ZHAO is unmissable.
Chapter 6 - Insurrection at Jiange (Mirror Realm) - BAO SANNIANG must be defeated at the very end of this mission. This one is unmissable.
Chapter 8 - The Battle against Tamamo (Mirror Realm) - LU BU and NEZHA must be defeated towards the end of the mission. These two are unmissable.