There aren't any. Hausdorff spaces are sober spaces. If $X, Y$ are sober, then every locale map $\mathcal{O}(Y) \to \mathcal{O}(X)$, i.e., every poset map between their topologies that preserves finite meets and arbitrary joins, arises from a uniquely determined continuous map $X \to Y$. It follows that a poset isomorphism $\mathcal{O}(X) \cong \mathcal{O}(Y)$, being a locale isomorphism, arises from a homeomorphism between the spaces.

(Just to give slightly more detail: for a sober space $X$, the points of $X$ are in natural bijection with locale maps $\mathcal{O}(X) \to \mathcal{O}(1)$ where the codomain is the topology on a one-point space. Thus a locale map $\phi: \mathcal{O}(Y) \to \mathcal{O}(X)$ induces, via composition with locale maps $\mathcal{O}(X) \to \mathcal{O}(1)$, a function $f: X \to Y$, and is itself of the form $\phi(V) = f^{-1}(V)$.)

There aren't any. Hausdorff spaces are sober spaces. If $X, Y$ are sober, then every frame map $\mathcal{O}(Y) \to \mathcal{O}(X)$, i.e., every poset map between their topologies that preserves finite meets and arbitrary joins, arises from a uniquely determined continuous map $X \to Y$. It follows that a poset isomorphism $\mathcal{O}(X) \cong \mathcal{O}(Y)$, being a frame isomorphism, arises from a homeomorphism between the spaces.

(Just to give slightly more detail: for a sober space $X$, the points of $X$ are in natural bijection with frame maps $\mathcal{O}(X) \to \mathcal{O}(1)$ where the codomain is the topology on a one-point space. Thus a frame map $\phi: \mathcal{O}(Y) \to \mathcal{O}(X)$ induces, via composition with frame maps $\mathcal{O}(X) \to \mathcal{O}(1)$, a function $f: X \to Y$, and is itself of the form $\phi(V) = f^{-1}(V)$.)

There aren't any. Hausdorff spaces are sober spaces. If $X, Y$ are sober, then every locale map $\mathcal{O}(Y) \to \mathcal{O}(X)$, i.e., every poset map between their topologies that preserves finite meets and arbitrary joins, arises from a uniquely determined continuous map $X \to Y$. It follows that a poset isomorphism $\mathcal{O}(X) \cong \mathcal{O}(Y)$, being a locale isomorphism, arises from a homeomorphism between the spaces.

There aren't any. Hausdorff spaces are sober spaces. If $X, Y$ are sober, then every locale map $\mathcal{O}(Y) \to \mathcal{O}(X)$, i.e., every poset map between their topologies that preserves finite meets and arbitrary joins, arises from a uniquely determined continuous map $X \to Y$. It follows that a poset isomorphism $\mathcal{O}(X) \cong \mathcal{O}(Y)$, being a locale isomorphism, arises from a homeomorphism between the spaces.

(Just to give slightly more detail: for a sober space $X$, the points of $X$ are in natural bijection with locale maps $\mathcal{O}(X) \to \mathcal{O}(1)$ where the codomain is the topology on a one-point space. Thus a locale map $\phi: \mathcal{O}(Y) \to \mathcal{O}(X)$ induces, via composition with locale maps $\mathcal{O}(X) \to \mathcal{O}(1)$, a function $f: X \to Y$, and is itself of the form $\phi(V) = f^{-1}(V)$.)