It is easy to see that the posets $(U,\le)$ that can arise in this way are exactly such that for all $u$ and $v$ in $U$ $$u\le v\ \&\ v\le u\implies u=v. \tag{1}\label{1}$$

Indeed, if for some $A$ and $B$ in $|R|/_{\sim}$ we have $A\le_R B\le_R A$, then for some $a,a_1$ in $A$ and some $b,b_1$ in $B$ we have $$aRb \quad \text{and}\quad bRb_1\!Ra_1\!Ra,$$ so that $aRb$ and, by the transitivity, $bRa$, so that $a\sim b$. So, $A\le_R B\le_R A$ implies $A=B$.

Vice versa, if \eqref{1} holds for a poset $(U,\le)$, let $R$ be $\le$. Then for any $u$ and $v$ in $U$ we have $u\sim v$ iff $u=v$ and hence $(|R|/_{\sim},\le_R)$ coincides with $(U,\le)$.

It is easy to see that the posets $(U,\le)$ that can arise in this way are exactly such that for all $u$ and $v$ in $U$ $$u\le v\ \&\ v\le u\implies u=v. \tag{1}\label{1}$$ That is, any poset can arise this way.

Indeed, if for some $A$ and $B$ in $|R|/_{\sim}$ we have $A\le_R B\le_R A$, then for some $a,a_1$ in $A$ and some $b,b_1$ in $B$ we have $$aRb \quad \text{and}\quad bRb_1\!Ra_1\!Ra,$$ so that $aRb$ and, by the transitivity, $bRa$, so that $a\sim b$. So, $A\le_R B\le_R A$ implies $A=B$.

Vice versa, if \eqref{1} holds for a poset $(U,\le)$, let $R$ be $\le$. Then for any $u$ and $v$ in $U$ we have $u\sim v$ iff $u=v$ and hence $(|R|/_{\sim},\le_R)$ coincides with $(U,\le)$.

The conditions that $X$ is a compact metric space and $R$ is closed play no role here.

It is easy to see that the posets $(U,\le)$ that can arise in this way are exactly such that for all $u$ and $v$ in $U$ $$u\le v\ \&\ v\le u\implies u=v.$$

It is easy to see that the posets $(U,\le)$ that can arise in this way are exactly such that for all $u$ and $v$ in $U$ $$u\le v\ \&\ v\le u\implies u=v. \tag{1}\label{1}$$

Indeed, if for some $A$ and $B$ in $|R|/_{\sim}$ we have $A\le_R B\le_R A$, then for some $a,a_1$ in $A$ and some $b,b_1$ in $B$ we have $$aRb \quad \text{and}\quad bRb_1\!Ra_1\!Ra,$$ so that $aRb$ and, by the transitivity, $bRa$, so that $a\sim b$. So, $A\le_R B\le_R A$ implies $A=B$.

Vice versa, if \eqref{1} holds for a poset $(U,\le)$, let $R$ be $\le$. Then for any $u$ and $v$ in $U$ we have $u\sim v$ iff $u=v$ and hence $(|R|/_{\sim},\le_R)$ coincides with $(U,\le)$.