Comment by generationP

Gröbner bases appear in the standard reference on real algebraic geometry ( https://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-po... ), so the answer may well be "yes", but don't ask me for the details. (All I can tell is that quadratic systems are no easier than degree-n systems in general -- there's a known trick how any system can be reduced to degree 2 by introduced extra variables.)