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It is (relatively) consistent with ZFC that this is false. For example, suppose that there are $\alpha<\beta$ with $L_\alpha$ and $L_\beta$ both being models of ZFC. (This is a very weak large cardinal axiom). Let us take $\alpha$ and $\beta$ to be least. So $L_\beta$ thinks that there is only one height $\alpha$ of a model of ZFC. It follows that $\alpha$ is countable inside $L_\beta$, and so inside $L_\beta$, the hyperverse is not empty---it contains $L_\alpha$ and many forcing extensions of $L_\alpha$ and so on. Let $x$ be a real in $L_\beta$ coding a relation on $\omega$ of order-type $\alpha$. This real cannot be in any CTM of ZFC inside $L_\beta$, since that model would have to have height at least $\beta$ by the minimality of $\beta$, and so it could not be an element of $L_\beta$.

Another way to argue: let $M_0$ be any countable transitive model of ZFC, and let $M$ be any $\in$-minimal countable transitive model of ZFC with $M_0\in M$. So $M_0$ is countable in $M$, but by minimality, $M$ can have no CTM containing a real coding $M_0$.

Meanwhile, if there is an inaccessible cardinal $\kappa$, or merely an uncountable worldly cardinal, then every real is contained in the corresponding $V_\kappa$, and by Löwenheim-Skolem, we may collapse a countable elementary substructure to place it into a countable transitive model of ZFC.

It is (relatively) consistent with ZFC that this is false. For example, suppose that there are $\alpha<\beta$ with $L_\alpha$ and $L_\beta$ both being models of ZFC. (This is a very weak large cardinal axiom). Let us take $\alpha$ and $\beta$ to be least. So $L_\beta$ thinks that there is only one height $\alpha$ of a model of ZFC. It follows that $\alpha$ is countable inside $L_\beta$, and so inside $L_\beta$, the hyperverse is not empty---it contains $L_\alpha$ and many forcing extensions of $L_\alpha$ and so on. Let $x$ be a real in $L_\beta$ coding a relation on $\omega$ of order-type $\alpha$. This real cannot be in any CTM of ZFC inside $L_\beta$, since that model would have to have height at least $\beta$ by the minimality of $\beta$, and so it could not be an element of $L_\beta$.

Another way to argue: let $M_0$ be any countable transitive model of ZFC, and let $M$ be any $\in$-minimal countable transitive model of ZFC with $M_0\in M$. So $M_0$ is countable in $M$, but by minimality, $M$ can have no CTM containing a real coding $M_0$.

Meanwhile, if there is an inaccessible cardinal $\kappa$, or merely a worldly cardinal, then every real is contained in the corresponding $V_\kappa$, and by Löwenheim-Skolem, we may collapse a countable elementary substructure to place it into a countable transitive model of ZFC.

It is (relatively) consistent with ZFC that this is false. For example, suppose that there are $\alpha<\beta$ with $L_\alpha$ and $L_\beta$ both being models of ZFC. (This is a very weak large cardinal axiom). Let us take $\alpha$ and $\beta$ to be least. So $L_\beta$ thinks that there is only one height $\alpha$ of a model of ZFC. It follows that $\alpha$ is countable inside $L_\beta$, and so inside $L_\beta$, the hyperverse is not empty. Let $x$ be a real in $L_\beta$ coding a relation on $\omega$ of order-type $\alpha$. This real cannot be in any CTM of ZFC inside $L_\beta$, since that model would have to have height at least $\beta$ by the minimality of $\beta$.

Another way to argue: let $M_0$ be any countable transitive model of ZFC, and let $M$ be any $\in$-minimal countable transitive model of ZFC with $M_0\in M$. So $M_0$ is countable in $M$, but by minimality, $M$ can have no CTM containing a real coding $M_0$.

It is (relatively) consistent with ZFC that this is false. For example, suppose that there are $\alpha<\beta$ with $L_\alpha$ and $L_\beta$ both being models of ZFC. (This is a very weak large cardinal axiom). Let us take $\alpha$ and $\beta$ to be least. So $L_\beta$ thinks that there is only one height $\alpha$ of a model of ZFC. It follows that $\alpha$ is countable inside $L_\beta$, and so inside $L_\beta$, the hyperverse is not empty---it contains $L_\alpha$ and many forcing extensions of $L_\alpha$ and so on. Let $x$ be a real in $L_\beta$ coding a relation on $\omega$ of order-type $\alpha$. This real cannot be in any CTM of ZFC inside $L_\beta$, since that model would have to have height at least $\beta$ by the minimality of $\beta$, and so it could not be an element of $L_\beta$.

Another way to argue: let $M_0$ be any countable transitive model of ZFC, and let $M$ be any $\in$-minimal countable transitive model of ZFC with $M_0\in M$. So $M_0$ is countable in $M$, but by minimality, $M$ can have no CTM containing a real coding $M_0$.

Meanwhile, if there is an inaccessible cardinal $\kappa$, or merely an uncountable worldly cardinal, then every real is contained in the corresponding $V_\kappa$, and by Löwenheim-Skolem, we may collapse a countable elementary substructure to place it into a countable transitive model of ZFC.

It is (relatively) consistent with ZFC that this is false. For example, suppose that there are $\alpha<\beta$ with $L_\alpha$ and $L_\beta$ both being models of ZFC. (This is a very weak large cardinal axiom). Let us take $\alpha$ and $\beta$ to be least. So $L_\beta$ thinks that there is only one height $\alpha$ of a model of ZFC. It follows that $\alpha$ is countable inside $L_\beta$, and so inside $L_\beta$, the hyperverse is not empty. Let $x$ be a real coding a relation on $\omega$ of order-type $\alpha$. This real cannot be in any CTM of ZFC inside $L_\beta$, since that model would have to have height at least $\beta$ by the minimality of $\beta$.

Another way to argue: let $M_0$ be any countable transitive model of ZFC, and let $M$ be any $\in$-minimal countable transitive model of ZFC with $M_0\in M$. So $M_0$ is countable in $M$, but by minimality, $M$ can have no CTM containing a real coding $M_0$.

It is (relatively) consistent with ZFC that this is false. For example, suppose that there are $\alpha<\beta$ with $L_\alpha$ and $L_\beta$ both being models of ZFC. (This is a very weak large cardinal axiom). Let us take $\alpha$ and $\beta$ to be least. So $L_\beta$ thinks that there is only one height $\alpha$ of a model of ZFC. It follows that $\alpha$ is countable inside $L_\beta$, and so inside $L_\beta$, the hyperverse is not empty. Let $x$ be a real in $L_\beta$ coding a relation on $\omega$ of order-type $\alpha$. This real cannot be in any CTM of ZFC inside $L_\beta$, since that model would have to have height at least $\beta$ by the minimality of $\beta$.

Another way to argue: let $M_0$ be any countable transitive model of ZFC, and let $M$ be any $\in$-minimal countable transitive model of ZFC with $M_0\in M$. So $M_0$ is countable in $M$, but by minimality, $M$ can have no CTM containing a real coding $M_0$.