Given two forcing notions $\mathbb{P}_0,\mathbb{P}_1$, we know that the product $\mathbb{P_0}\times\mathbb{P_1}$ will produce the following diagram of models ordered by inclusion:

where $M$ is the ground model, $G_i$ is a $\mathbb{P_i}$-generic filter over $M$ and $G_0\times G_1$ is $\mathbb{P_0}\times\mathbb{P_1}$-generic over $M$. Note that $M[G_0]\cap M[G_1] = M$.

My question is whether, given three forcing notions $\mathbb{P_0},\mathbb{P_1},\mathbb{P_2}$, we can define a forcing $\mathbb{Q}$ that induces the following lattice:

where, again, $G_i$ is $\mathbb{P_i}$-generic over $M$. I also would like $M[G_i]\cap M[G_j] = M$ for $i\neq j$. Note that, since the diagram represents a lattice, once we have $G_i$ and $G_j$ for some $i\neq j$, we are able to construct (modulo $M$) also $G_k$ with $k\neq i,j$.
Is such a construction known to be possible in a reasonably general (i.e. definable) way? Any reference?
Thanks

Given two forcing notions $\mathbb{P}_0,\mathbb{P}_1$, we know that the product $\mathbb{P_0}\times\mathbb{P_1}$ will produce the following diagram of models ordered by inclusion:

where $M$ is the ground model, $G_i$ is a $\mathbb{P_i}$-generic filter over $M$ and $G_0\times G_1$ is $\mathbb{P_0}\times\mathbb{P_1}$-generic over $M$. Note that $M[G_0]\cap M[G_1] = M$.

My question is whether, given three forcing notions $\mathbb{P_0},\mathbb{P_1},\mathbb{P_2}$, we can define a forcing $\mathbb{Q}$ that induces the following lattice:

where, again, $G_i$ is $\mathbb{P_i}$-generic over $M$. I also would like $M[G_i]\cap M[G_j] = M$ for $i\neq j$. Note that, since the diagram represents a lattice, once we have $G_i$ and $G_j$ for some $i\neq j$, we would be able to construct (modulo $M$) also $G_k$ with $k\neq i,j$.
Is such a construction known to be possible in a reasonably general (i.e. definable) way? Any reference?
Thanks