Define a complete embedding of Boolean algebra as an homomorphism of Boolean algebras which preserves also the sup and inf operations. Notice that if $\mathbb{B}$ and $\mathbb{D}$ are complete boolean algebras, $i:\mathbb{B}\to\mathbb{D}$ is a complete embedding and $G$ is $V$-generic for $\mathbb{D}$, then $H=i^{-1}[G]$ is $V$-generic for $\mathbb{B}$.

I'm curious to know if the following can be the case:

Assume $\mathbb{B}$ and $\mathbb{D}$ are complete stationary set preserving boolean algebras. Can there be two distinct complete embeddings $i_0:\mathbb{B}\to\mathbb{D}$, $i_1:\mathbb{B}\to\mathbb{D}$ such that if $G$ is $V$-generic for $\mathbb{D}$ and $H_j=i_j^{-1}[G]$ are the corresponding $V$-generic filters for $\mathbb{B}$ induced by the respective $i_j$, we can have that there is a name $\tau$ in the forcing language for $\mathbb{B}$ such that:

$\|\tau$ is a stationary subset of $\omega_1\|=1_{\mathbb{B}}$

$V[G]\models\sigma_{H_0}(\tau)$ is a stationary subset of $\omega_1$

$V[G]\models\sigma_{H_1}(\tau)$ is non-stationary