  We prove that the cycle-valued logarithmic Gromov--Witten theory of a product
of simple normal crossings pairs $X\times Y$ decomposes into a product of
pieces coming from $X$ and $Y$, provided that the decomposition is considered
over a blowup of the moduli space of curves. The result is established by
applying the toroidal semistable reduction theorem to the stabilization
morphism on the stack of maps to the Artin fans. Since products of smooth pairs
are naturally simple normal crossings pairs, the result provides a direct
avenue of access to questions in logarithmic Gromov--Witten theory via relative
Gromov--Witten theory.
