Existential Theory of the Reals in NP?
No, unfortunately that doesn't work.  The certificate needs to be a sequence of bits that is verifiable in polynomial time -- and thus the length of the certificate has to be at most a polynomial in the length of the input instance.  If your certificate is the value of some variables, and those values are real numbers, you might not be able to write down their binary representation in any reasonable amount of space (indeed, their binary representation might be infinite-length).

For instance, consider the sentence $\exists x . x^2 = 2$.  This sentence is valid, as $x= \sqrt{2}$ is an example that makes $x^2=2$ true.  However, when you try to write out $\sqrt{2}$ in binary, you get an infinite sequence, so it's not clear what the short certificate would be.  (Certificates need to be of finite length, certainly.)  In general, representing real numbers in binary is messy business; no matter what representation you choose, there will always be some real numbers you can't represent finitely.