Before the War
The Leones began to have bad terms with the Triads sometime during the events of Grand Theft Auto: Liberty City Stories when the Triads blew up a Leone warehouse (which then became the Turtle Head Fish Factory, a front for the Triads). They were trying to push out the Leones for the Sicilian Mob. Later on in 2001, during the mission Farewell 'Chunky' Lee Chong, Joey Leone sent Claude to kill Lee Chong for selling cheap SPANK in his noodle stand.
The War Begins
In the mission Cipriani's Chauffeur, Toni Cipriani tells Claude to drive him to Mr. Wong's Laundromat which pays protection to the Leones but only to find out that they now pay protection to the Triads. In the next mission, Taking Out The Laundry, Cipriani tells Claude to blow up three Triad Fish Vans. Cipriani then tells Claude that Mr. Wong agreed to pay protection and tells him to pick up the money, but when Claude gets there, he is ambushed by the Triads. Claude manages to kill them all and take the money back to Toni. In the mission Salvatore's Called A Meeting, Claude picks up Joey Leone, Luigi Goterelli and Toni Cipriani but during the way they are attacked by the Triads, once again. Claude manages to escape from them all off and gets to Salvatore's Gentlemen's Club where Leone Don Salvatore Leone discusses about the Triads.
Ending of the War
After a while, Toni instructs Claude and three other Leone soldiers to kill three Triad warlords in which they successfully managed to pull off. During the mission, whilst driving around Chinatown and Saint Mark's, the player will see a large shootout between Leone and Triad gangsters. Finally in, the mission Blow Fish, Toni tells Claude to get a dustcart rigged with a bomb set up by 8-Ball and is told to bring it to the Turtle Head Fish Factory and blow it up. Although the Triads can still be seen in Chinatown after these events, they are never mentioned again in the storyline, suggesting that the Leones won the war.