Several researchers have recently established that for every Turing degree c, the real closed field of all c-computable real numbers has spectrum {d : d ′ ≥ c ′′ }. We investigate the spectra of real closed fields further, focusing first on subfields of the field R0 of computable real numbers, then on archimedean real closed fields more generally, and finally on nonarchimedean real closed fields. For each noncomputable, computably enumerable set C, we produce a real closed C-computable subfield of R0 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. * This is a pre-print of an article to be published in the Archive for Mathematical Logic. The final authenticated version will be available online at: https://doi.org/10.1007/s00153-018-0638-z.