Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the answer to the question have an effect on the PSPACE proof by J. Canny? For example, ETR is not in PSPACE, if numbers in real closed field are allowed as coefficients?

In order for this to be a computational problem in the first place, you have to fix a representation of the coefficients by finite strings (which in particular implies that the field is countable). The answer will in general depend on the representation. For the most obvious case, if the coefficients are taken from the field of real algebraic numbers, and are represented in a common way (minimal polynomial + an isolating interval or a BKR sign condition), then the problem is equivalent to the one with rational coefficients, because we can just plug the definitions of the coefficients into the formula. 

