  In the product $L_1\times L_2$ of two Kripke complete consistent logics,
local tabularity of $L_1$ and $L_2$ is necessary for local tabularity of
$L_1\times L_2$. However, it is not sufficient: the product of two locally
tabular logics can be not locally tabular. We provide extra semantic and
axiomatic conditions which give criteria of local tabularity of the product of
two locally tabular logics. Then we apply them to identify new families of
locally tabular products.
