Since the linear parts $\langle Lx-bz\rangle$ and $\langle Cy\rangle$ have disjoint sets of variables, we can tropicalize them separately in $\mathbb{R}^{n+1}$ and $\mathbb{R}^m$, add appropriate lineality spaces, and then stably intersect them. This should speed up the computation of the tropical linear space.
Since the linear parts$\langle Lx-bz\rangle$ and $\langle Cy\rangle$ have disjoint sets of variables, we can tropicalize them separately in $\mathbb{R}^{n+1}$ and $\mathbb{R}^m$ , add appropriate lineality spaces, and then stably intersect them. This should speed up the computation of the tropical linear space.