  Several researchers have recently established that for every Turing degree
$\boldsymbol{c}$, the real closed field of all $\boldsymbol{c}$-computable real
numbers has spectrum $\{\boldsymbol{d}~:~\boldsymbol{d}'\geq\boldsymbol{c}"\}$.
We investigate the spectra of real closed fields further, focusing first on
subfields of the field $\mathbb{R}_{\boldsymbol{0}}$ of computable real
numbers, then on archimedean real closed fields more generally, and finally on
non-archimedean real closed fields. For each noncomputable, computably
enumerable set $C$, we produce a real closed $C$-computable subfield of
$\mathbb{R}_{\boldsymbol{0}}$ with no computable copy. Then we build an
archimedean real closed field with no computable copy but with a computable
enumeration of the Dedekind cuts it realizes, and a computably presentable
nonarchimedean real closed field whose residue field has no computable
presentation.
