In the unbounded-error case, it is known that both realtime quantum and probabilistic finite automata can recognize some uncomputable languages if they are allowed to use arbitrary real numbers in their transitions (Rabin, 1963; Yakaryilmaz and Say, 2011).
In the bounded-error case, we have a similar result for poly-time quantum Turing machines as well, i.e. the cardinality of $ \mathsf{BQP}_{\mathbb{C}} $ is uncountable (Adleman et. al., 1997).
My question is whether any bounded-error probabilistic space (time) class defined with unrestricted real numbers contains an uncomputable (or a recursive enumerable) language.
It is also known that (Watrous, 2003) if we restrict ourselves to algebraic numbers ($\mathbb{A}$), for any space-constructable function $ s(n) \in \Omega(\log n) $, \begin{equation} \mathsf{PrQSPACE}_{\mathbb{A}}(s) \subseteq \mathsf{DSPACE}(s^2), \end{equation} where $\mathsf{PrQSPACE}$ stand for unbounded-error quantum space.
Any partial answer (for the case of bounded-error probabilistic computation using non-algebraic transitions) violating this upper bound would also be nice.