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Let $M$ and $N$ be two unital JB$^*$-algebras and let $\mathcal{U} (M)$ and$\mathcal{U} (N)$ denote the sets of all unitaries in $M$ and $N$,respectively. We prove that the following statements are equivalent:  $(a)$ $M$ and $N$ are isometrically isomorphic as (complex) Banach spaces;  $(b)$ $M$ and $N$ are isometrically isomorphic as real Banach spaces;  $(c)$ There exists a surjective isometry $\Delta: \mathcal{U}(M)\to\mathcal{U}(N).$  We actually establish a more general statement asserting that, under somemild extra conditions, for each surjective isometry $\Delta:\mathcal{U} (M) \to\mathcal{U} (N)$ we can find a surjective real linear isometry $\Psi:M\to N$which coincides with $\Delta$ on the subset $e^{i M_{sa}}$. If we assume that$M$ and $N$ are JBW$^*$-algebras, then every surjective isometry$\Delta:\mathcal{U} (M) \to \mathcal{U} (N)$ admits a (unique) extension to asurjective real linear isometry from $M$ onto $N$. This is an extension of theHatori--Moln{\'a}r theorem to the setting of JB$^*$-algebras.

Can one identify two unital JB$^*$-algebras by the metric spaces determined by their sets of unitaries?