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We provide a real algebraic symbolic-numeric algorithm for computing the real variety V"R(I) of an ideal I@?R[x], assuming V"R(I) is finite (while V"C(I) could be infinite). Our approach uses sets of linear functionals on R[x], vanishing on a given set of polynomials generating I and their prolongations up to a given degree, as well as on polynomials of the real radical ideal IR obtained from the kernel of a suitably defined moment matrix assumed to be positive semidefinite and of maximum rank. We formulate a condition on the dimensions of projections of these sets of linear functionals, which serves as a stopping criterion for our algorithm; this new criterion is satisfied earlier than the previously used stopping criterion based on a rank condition for moment matrices. This algorithm is based on standard numerical linear algebra routines and semidefinite optimization and combines techniques from previous work of the authors together with an existing algorithm for the complex variety.