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Let me give the condensed perspective: Regarding $A$ as a discrete condensed ring, I think the structure of the "internal spectrum" is codified by the functor that takes any extremally disconnected profinite set $S$ to the poset of sheaves of prime ideals in the constant sheaf on $A$ over $S$. (One could forget the poset structure and regard it only as a condensed set. I will comment below what structure this remembers.) Here, a "sheaf of prime ideals" is defined to be a sheaf of ideals $I$ of $A$ together with a sheaf of multiplicative subsets $M$ of $A$ such that the map $I\sqcup M\to A$ is an isomorphism (of sheaves of sets); I hope this is the correct way to talk about "internal prime ideals"?

I claim that this is the "correct" answer to this question. Recall that Makkai's conceptual completeness theorem as explained by Lurie in his course on categorical logic, or by Barwick-Haine in their work on exodromy, gives a fully faithful embedding of the category of coherent topoi into the category of condensed categories; it takes coherent locales (aka spectral spaces) to condensed posets. In one direction, this takes any coherent topos to the condensed category of points. (This explains the name "conceptual completeness", as a strong version of Gödel's/Deligne's completeness theorem, that everything is determined by models, i.e. points.)

Summary: The spectrum of a ring is naturally a spectral space, i.e. coherent locale, so determined by its condensed poset of points. This is precisely the spectrum of $A$ as constructed internally in condensed sets.

Addendum: If one forgets the poset structure and only looks at the condensed set of prime ideals, one actually ends up getting a condensed set that is representable by a profinite set, which is precisely $\mathrm{Spec}(A)$ with its constructible topology.

Let me give the condensed perspective: Regarding $A$ as a discrete condensed ring, I think the structure of the "internal spectrum" is codified by the functor that takes any extremally disconnected profinite set $S$ to the poset of sheaves of prime ideals in the constant sheaf on $A$ over $S$. (One could forget the poset structure and regard it only as a condensed set. I will comment below what structure this remembers.) Here, a "sheaf of prime ideals" is defined to be a sheaf of ideals $I$ of $A$ together with a sheaf of multiplicative subsets $M$ of $A$ such that the map $I\sqcup M\to A$ is an isomorphism (of sheaves of sets); I hope this is the correct way to talk about "internal prime ideals"?

I claim that this is the "correct" answer to this question. Recall that Makkai's conceptual completeness theorem as explained by Lurie in his course on categorical logic, or by Barwick-Glasman-Haine in their work on exodromy, gives a fully faithful embedding of the category of coherent locales (aka spectral spaces) to condensed posets. In one direction, this takes any coherent locale to the condensed category of points.

Summary: The spectrum of a ring is naturally a spectral space, i.e. coherent locale, so determined by its condensed poset of points. This is precisely the spectrum of $A$ as constructed internally in condensed sets.

Addendum: If one forgets the poset structure and only looks at the condensed set of prime ideals, one actually ends up getting a condensed set that is representable by a profinite set, which is precisely $\mathrm{Spec}(A)$ with its constructible topology.

Let me give the condensed perspective: Regarding $A$ as a discrete condensed ring, I think the structure of the "internal spectrum" is codified by the functor that takes any extremally disconnected profinite set $S$ to the poset of sheaves of prime ideals in the constant sheaf on $A$ over $S$. (One could forget the poset structure and regard it only as a condensed set. I will comment below what structure this remembers.)

I claim that this is the "correct" answer to this question. Recall that Makkai's conceptual completeness theorem as explained by Lurie in his course on categorical logic, or by Barwick-Haine in their work on exodromy, gives a fully faithful embedding of the category of coherent topoi into the category of condensed categories; it takes coherent locales (aka spectral spaces) to condensed posets. In one direction, this takes any coherent topos to the condensed category of points. (This explains the name "conceptual completeness", as a strong version of Gödel's/Deligne's completeness theorem, that everything is determined by models, i.e. points.)

Summary: The spectrum of a ring is naturally a spectral space, i.e. coherent locale, so determined by its condensed poset of points. This is precisely the spectrum of $A$ as constructed internally in condensed sets.

Addendum: If one forgets the poset structure and only looks at the condensed set of prime ideals, one actually ends up getting a condensed set that is representable by a profinite set, which is precisely $\mathrm{Spec}(A)$ with its constructible topology.

Let me give the condensed perspective: Regarding $A$ as a discrete condensed ring, I think the structure of the "internal spectrum" is codified by the functor that takes any extremally disconnected profinite set $S$ to the poset of sheaves of prime ideals in the constant sheaf on $A$ over $S$. (One could forget the poset structure and regard it only as a condensed set. I will comment below what structure this remembers.) Here, a "sheaf of prime ideals" is defined to be a sheaf of ideals $I$ of $A$ together with a sheaf of multiplicative subsets $M$ of $A$ such that the map $I\sqcup M\to A$ is an isomorphism (of sheaves of sets); I hope this is the correct way to talk about "internal prime ideals"?

I claim that this is the "correct" answer to this question. Recall that Makkai's conceptual completeness theorem as explained by Lurie in his course on categorical logic, or by Barwick-Haine in their work on exodromy, gives a fully faithful embedding of the category of coherent topoi into the category of condensed categories; it takes coherent locales (aka spectral spaces) to condensed posets. In one direction, this takes any coherent topos to the condensed category of points. (This explains the name "conceptual completeness", as a strong version of Gödel's/Deligne's completeness theorem, that everything is determined by models, i.e. points.)

Summary: The spectrum of a ring is naturally a spectral space, i.e. coherent locale, so determined by its condensed poset of points. This is precisely the spectrum of $A$ as constructed internally in condensed sets.

Addendum: If one forgets the poset structure and only looks at the condensed set of prime ideals, one actually ends up getting a condensed set that is representable by a profinite set, which is precisely $\mathrm{Spec}(A)$ with its constructible topology.

Let me give the condensed perspective: Regarding $A$ as a discrete condensed ring, I think the structure of the "internal spectrum" is codified by the functor that takes any extremally disconnected set $S$ to the frame of open subsets of $\mathrm{Spec}(\mathrm{Cont}(S,A))$, i.e. one gets a condensed frame (thus, a certain kind of condensed poset).

I claim that this is the "correct" answer to this question. Recall that Makkai's conceptual completeness theorem as explained by Lurie in his course on categorical logic, or by Barwick-Haine in their work on exodromy, gives a fully faithful embedding of the category of coherent topoi into the category of condensed categories; it takes coherent locales (aka spectral spaces) to condensed posets. In one direction, this takes any coherent topos to the condensed category of points. (This explains the name "conceptual completeness", as a strong version of Gödel's/Deligne's completeness theorem, that everything is determined by models, i.e. points.)

Summary: The spectrum of a ring is naturally a spectral space, i.e. coherent locale, so determined by its condensed poset of points. This is precisely the spectrum of $A$ as constructed internally in condensed sets.

Let me give the condensed perspective: Regarding $A$ as a discrete condensed ring, I think the structure of the "internal spectrum" is codified by the functor that takes any extremally disconnected profinite set $S$ to the poset of sheaves of prime ideals in the constant sheaf on $A$ over $S$. (One could forget the poset structure and regard it only as a condensed set. I will comment below what structure this remembers.)

I claim that this is the "correct" answer to this question. Recall that Makkai's conceptual completeness theorem as explained by Lurie in his course on categorical logic, or by Barwick-Haine in their work on exodromy, gives a fully faithful embedding of the category of coherent topoi into the category of condensed categories; it takes coherent locales (aka spectral spaces) to condensed posets. In one direction, this takes any coherent topos to the condensed category of points. (This explains the name "conceptual completeness", as a strong version of Gödel's/Deligne's completeness theorem, that everything is determined by models, i.e. points.)

Summary: The spectrum of a ring is naturally a spectral space, i.e. coherent locale, so determined by its condensed poset of points. This is precisely the spectrum of $A$ as constructed internally in condensed sets.

Addendum: If one forgets the poset structure and only looks at the condensed set of prime ideals, one actually ends up getting a condensed set that is representable by a profinite set, which is precisely $\mathrm{Spec}(A)$ with its constructible topology.