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My recollection is that Macintyre proved there are $2^{\aleph_0}$ complete theories $(K,L)$ where $L\subset K$ are real closed fields. This is in his thesis but I don't think he published it anywhere else. There are later papers of Francoise Delon and Walter Bauer that develop this further. On the other hand there is an earlier theorem of Robinson's that if $L\subset K$ are real closed and $L$ is dense in $K$ then the theory of $(K,L)$ is decidable.