The real spectra of certain "twisted" polynomial algebrasAover R are examined. In certain cases, including Weyl algebras and universal enveloping algebras of solvable Lie algebras, the stability indices of the residue spaces of Sper(A) are estimated. The results show that the Bröcker-Scheiderer theory of minimal generation of constructible sets applies to Sper(A) in these cases. An attempt is made to compute all orderings on the Weyl algebra over R. Several open problems are mentioned.