The martial art practiced by only the hardiest and most worthy Fools and Clowns. This has a long history pre-dating the formal establishment of Fools' Guilds in Quirm and Ankh-Morpork. Anyone doubting this should reflect that clowning is, at bottom, a carefully modulated and choreographed process of insane violence, where nobody gets hurt. (Physically hurt, at least). Sloshi is essentially clowning without the moderation.
It has its origins in the hinterlands of Überwald many years ago, where travelling companies of clowns would fight for the choicer sites. (While the The New Discworld Companion does not specifically say this, anyone travelling in Überwald has other perils to contend with, such as vampires, werewolves and feral banshees and must need some sort of a defence against them...). Eventually, this became a form of formal duelling among Überwaldean student clowns in the town of Müning, where the scars of a sloshi fighter were worn with pride. (but only in very close company, as while sloshi fighting involved a measure of protective clothing, in Müning the buttocks were left bare. This also became the noun for the act of demonstrating one's sloshi scars - müning. A similar basic thought process is thought to have been responsible for the name of the Quirmian clowning centre at La Sorbumme.
It is on record that the Fools' Guild has provided a Regiment in the past that has marched - memorably - to the defence of Ankh-Morpork. This would have included many clowns well-versed in sloshi techniques. In fact, a seventh-nose Dan and sloshi-sensai, such as Uncle Bootsie, was capable of slaying seventeen Pseudopolitian mercenaries in a single mêlée, armed only with a ladder and two buckets of wallpaper paste. It is said, although only by clowns, that when this was first demonstrated on the field of combat, forty-one other mercaneries were incapacitated by laughter and overpowered by the rest of the Clowns' Battallion in addition to the seventeen slain by the ladder and the paste.
Mavolio Bent was only resurrecting this tradition when he slew two Assassins with a pink balloon and incapacitated several onlookers at Moist von Lipwig's trial - with, indeed, the traditional ladder closely followed by eight custard pies in the air at once.
"Sloshed" is used figuratively in English to refer to someone being drunk. In this context, Sloshi could be read as a Discworld analogue to the Chinese martial art of Drunken boxing, a styles of Chinese martial arts that imitate the movements of a drunk person.
The scars associated with Sloshi (as well as the practice of keeping the buttocks bare) is likely a reference to the of Academic Fencing practiced by some students in Central European universities. Academic fencers would fight without facial protection, with the resulting duelling scars being was as a mark of their class and honour, due to the status of duelling societies at German and Austrian universities at the time.
Annotation for non-English speakers
Sloshi is a homonym for sloshy. The verb to slosh means To spill or splash (a liquid) copiously or clumsily: for example, to slosh paint on the floor. Or indeed across another clown's face.