Orbit equivalence, coinduced actions and free products

The following result is proven. Let $G_1 \cc^{T_1} (X_1,\mu_1)$ and $G_2 \cc^{T_2} (X_2,\mu_2)$ be orbit-equivalent, essentially free, probability measure preserving actions of countable groups $G_1$ and $G_2$. Let $H$ be any countable group. For $i=1,2$, let $\Gamma_i = G_i *H$ be the free product. Then the actions of $\Gamma_1$ and $\Gamma_2$ coinduced from $T_1$ and $T_2$ are orbit-equivalent. As an application, it is shown that if $\Gamma$ is a free group, then all nontrivial Bernoulli shifts over $\Gamma$ are orbit-equivalent.