Solidity of type III Bernoulli crossed products

We generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra $A_0$, any faithful normal state $\varphi_0$ and any discrete group $\Gamma$, the associated Bernoulli crossed product von Neumann algebra $M=(A_0,\varphi_0)^{\mathbin{\bar{\otimes}}\Gamma}\rtimes \Gamma$ is solid relatively to $\mathcal{L}(\Gamma)$. In particular, if $\mathcal{L}(\Gamma)$ is solid then $M$ is solid and if $\Gamma$ is non-amenable and $A_0 \neq \mathbb{C}$ then $M$ is a full prime factor. This gives many new examples of solid or prime type $\mathrm{III}$ factors. Following Chifan and Ioana, we also obtain the first examples of solid non-amenable type $\mathrm{III}$ equivalence relations.