First of all, thank you for the adjective "fantastic"(!). The question is actually studied in a paper of mine [(link to the MR review)][1]. Sorry for talking about my own paper (I have no option, since I do not know if anyone else is interested enough in these questions). 

[I should add that, in all this, $g\geq 2$].
In the paper referred to above, what is proved is that any intermediate subgroup either has finite index in $Sp_{2g}(O_K)$ or else contains $Sp_{2g}(\mathbb{Z})$ as a finite index subgroup. In particular, since $Sp_{2g}(\mathbb{Z})$ is a maximal discrete subgroup of $Sp_{2g}(\mathbb{R})$ by the answers to your previous  question, there are no in-between subgroups of infinite index in $Sp_{2g}(O_K)$ other than $Sp_{2g}(\mathbb{Z})$.  


  [1]: http://www.ams.org/mathscinet-getitem?mr=1324463