A constrained support can make single-site Gibbs reducible

The coordinate marginals are uniform on $\{0,1\}$. For coordinate 1 set $K_{\{1\}}(t,\cdot )={\delta }_t$; for any measurable $C\subseteq S$ and $D\subseteq \{0,1\}$, $\pi (C\cap p_2^{-1}(D))=\frac{1}{2}{\sum }_{t\in D}{\mathbf{1}}_C(t,t)={\int }_D{K}_{\{1\}}(t,C_t){\pi }_2(dt)$, where $C_t=\{s:(s,t)\in C\}$. Exchanging the coordinates gives the same identity for $K_{\{2\}}(t,\cdot )={\delta }_t$. Each section is a probability measure at every conditioning value, and the identities for all $C,D$ prove that these are proper full conditional kernels by [F3].

Starting from either state $(t,t)\in E$, updating coordinate 1 draws $t$ from $K_{\{1\}}(t,\cdot )$, and updating coordinate 2 draws $t$ from $K_{\{2\}}(t,\cdot )$. Thus both component transition matrices restricted to $E$ are the identity. Each singleton $\{(0,0)\}$ and $\{(1,1)\}$ is nonempty, proper, and closed, so [F2] makes the restricted finite chain reducible.