  Let $(S,\mathfrak{m},k)$ and $(T,\mathfrak{n},k)$ be local rings, and let $R$
denote their fiber product over their common residue field $k$. Inspired by
work of Naseh and Sather-Wagstaff, we explore consequences of vanishing of
${\rm Tor}^R_m(M,N)$ for various values of $m$, where $M$ and $N$ are finitely
generated $R$-modules.
