The answer is no. Indeed, assume that such sets $A,B,C$ exist. Write $a*b=ab$ for brevity. Let $a \in A$, $b \in B$, $c \in C$. We first show that $ba \in C$. If not, there are two cases: (i) $ba \in A$. Then $(ba)b \in C$, but also $b(ab) \in A$, since $ab \in C$. Contradiction. (ii) $ba \in B$. Then $a(ba) \in C$, but also $(ab)a \in B$, since $ab \in C$. Contradiction. So $ba \in C$, and similarly $cb \in A$, $ac \in B$. This shows that $a^2b = a(ab) \in B$, which is only possible if $a^2 \in B$. But then $a^2c \in A$. However at the same time $a^2c=a(ac) \in C$, since $ac \in B$, which is absurd.