Vanishing of relative homology and depth of tensor products

For finitely generated modules $M$ and $N$ over a Gorenstein local ring $R$, one has $depthM+depthN=depth(M\otimes N)+depthR$, i.e., the depth formula holds, if $M$ and $N$ are Tor-independent and Tate homology ${\widehat{Tor}}_i^R(M,N)$ vanishes for all $i\in \mathbb{Z}$. We establish the same conclusion under weaker hypotheses: if $M$ and $N$ are $\mathcal{G}$-relative Tor-independent, then the vanishing of ${\widehat{Tor}}_i^R(M,N)$ for all $i\le 0$ is enough for the depth formula to hold. We also analyze the depth of tensor products of modules and obtain a necessary condition for the depth formula in terms of $\mathcal{G}$-relative homology.