Anatoli is the brother of a more famous member of the Boisa family - Vladimer, who played in the Euroleague with Olimpija Ljubljana for several seasons. Anatoli lacks some skills that his brother has, but practice and hard work helped him have a career of his own. He started his early career in Rustavi, then left for the USA and played in the NCAA for a year. His next team was German side Mitteldeutscher BC Weissenfels. Boisa came back to Georgia and became a Superleague champion on several occasions. Boisa is not the best scorer, but he always excels on defence. That is the reason why he has been a member of both youth and senior national teams for many years and always finds a spot on the roster of top domestic teams.