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RCF is not aleph-zero-categorical: the real algebraic numbers and a countable real closure of $\mathbb{Q}(t)$ are countable models with different transcendence degrees. It is not categorical in any uncountable cardinal either. At cardinality $2^{{\aleph }_0}$, for example, $\mathbb{R}$ is Archimedean whereas the real closure of $\mathbb{R}(t)$ with $t$ infinitely large is non-Archimedean. If RCF were categorical in any uncountable cardinal, the Morley categoricity theorem would make it categorical in every uncountable cardinal, contradicting this pair.