  We show that $|MS(L_1 # L_2)|=|MS(L_1)|\times|MS(L_2)|\times\mathbb{R}$ when
$L_1$ and $L_2$ are any non-split and non-fibred links. Here $MS(L)$ denotes
the Kakimizu complex of a link $L$, which records the taut Seifert surfaces for
$L$. We also show that the analogous result holds if we study incompressible
Seifert surfaces instead of taut ones.
