*This was originally part of this older question of mine, but in retrospect that question should have been broken into two parts - this is the still-unanswered part.*

Let $\Sigma$ be the language of ordered fields and let $\mathcal{R}$ be the field of real numbers. Say that a theory $T$ in a language $\Sigma'$ gotten by adding constant symbols to $\Sigma$ is **$\mathcal{R}$-satisfiable** iff $T$ has a model whose $\Sigma$-reduct is $\mathcal{R}$ itself. For cardinals $\kappa<\lambda$, say that *$\mathcal{R}$-satisfiability is $(\kappa,\lambda)$-compact* iff every theory of cardinality $<\lambda$ all of whose size-$<\kappa$-subsets are $\mathcal{R}$-satisfiable is itself $\mathcal{R}$-satisfiable.

There are a couple easy observations about the possible extent of compactness for $\mathcal{R}$-satisfiability (see the discussion at the above-linked question):

$\mathcal{R}$-satisfiability is provably not $(\omega_1,\omega_2)$-compact or $(2^\omega, (2^\omega)^{+})$-compact.

If $\kappa$ is measurable then $\mathcal{R}$-satisfiability is $(\kappa,\kappa^+)$-compact.

However, there is still one natural "low-level" question that remains open:

Is $\mathcal{R}$-satisfiability consistently $(\omega_2,\omega_3)$-compact?

Here are a couple comments:

First, by the above observations $\mathcal{R}$-satisfiability can only be $(\omega_2,\omega_3)$-compact if $2^\omega\ge\omega_3$ (this was pointed out by Joel), so neither $\mathsf{CH}$ nor forcing axioms can help us.

A bit more complicatedly, there are subtleties which pose potential obstacles to any "coarse" argument. Specifically, suppose we shift from $\mathcal{R}$ to its expansion $\mathcal{R}_\mathbb{Z}$ by a predicate naming the integers. In $\mathcal{R}_\mathbb{Z}$ we can talk about reals coding countable well-orderings and comparisons between the ordertypes of coded well-orderings. This gives us right off the bat a $\mathsf{ZFC}$-provable counterexample to $(\omega_2,\omega_3)$-compactness of $\mathcal{R}_\mathbb{Z}$-satisfiability: let $T$ be the theory using constant symbols $(c_\eta)_{\eta<\omega_2}$ saying that the $c_\eta$s code well-orderings of distinct ordertypes. This $T$ also demonstrates that we cannot obviously focus on complete theories WLOG: while every subtheory of $T$ of size $\omega_1$ is $\mathcal{R}_\mathbb{Z}$-satisfiable, every completion $S$ of $T$ has some subtheory of size $\omega_1$ which is not $\mathcal{R}_\mathbb{Z}$-satisfiable since every linear order of size $\omega_2$ has a suborder of size $\omega_1$ not embedding into $\omega_1$. Of course, the above reasoning breaks down for $\mathcal{R}$, but this does still suggest that there may be subtleties (and ultimately makes me very skeptical of a positive answer).