  The real radical ideal of a system of polynomials with finitely many complex
roots is generated by a system of real polynomials having only real roots and
free of multiplicities. It is a central object in computational real algebraic
geometry and important as a preconditioner for numerical solvers. Lasserre and
co-workers have shown that the real radical ideal of real polynomial systems
with finitely many real solutions can be determined by a combination of
semi-definite programming (SDP) and geometric involution techniques. A
conjectured extension of such methods to positive dimensional polynomial
systems has been given recently by Ma, Wang and Zhi.
  We show that regularity in the form of the Slater constraint qualification
(strict feasibility) fails for the resulting SDP feasibility problems. Facial
reduction is then a popular technique whereby SDP problems that fail strict
feasibility can be regularized by projecting onto a face of the convex cone of
semi-definite problems.
  In this paper we introduce a framework for combining facial reduction with
such SDP methods for analyzing $0$ and positive dimensional real ideals of real
polynomial systems. The SDP methods are implemented in MATLAB and our geometric
involutive form is implemented in Maple. We use two approaches to find a
feasible moment matrix. We use an interior point method within the CVX package
for MATLAB and also the Douglas-Rachford (DR) projection-reflection method.
  Illustrative examples show the advantages of the DR approach for some
problems over standard interior point methods. We also see the advantage of
facial reduction both in regularizing the problem and also in reducing the
dimension of the moment matrices. Problems requiring more than one facial
reduction are also presented.
