The realization space is
  [1   0   1                x3   -x1*x2^3*x3 - x1*x2^3 + 2*x1*x2^2*x3 + x1*x2^2 - x1*x2*x3 + x2^2*x3 + x2*x3^2 - x2*x3 - x3^2    1   x1*x2^2 - x1*x2 - x3   0   x1*x2^2 - x1*x2 - x3    0    1]
  [0   1   1   x1*x2^2 - x1*x2                                                           -x1*x2^3 + x1*x2^2 + x2^2*x3 - x2*x3   x2                      0   0                    -x2    1   x2]
  [0   0   0                x3                                                                   x1*x2^2*x3 - x1*x2*x3 - x3^2    1                    -x3   1                    -x3   x1   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal (x1^2*x2^2 - x1^2*x2 + x1*x2 - x1*x3 - x3)
avoiding the zero loci of the polynomials
RingElem[x1*x2 - x3, x1, x1*x2 - x1 - x3, x1^2*x2^2 - x1^2*x2 - x1*x2*x3 + x3^2 - x3, x1^4*x2^5 - 3*x1^4*x2^4 + 3*x1^4*x2^3 - x1^4*x2^2 + x1^3*x2^5*x3 - 3*x1^3*x2^4*x3 + x1^3*x2^4 + 3*x1^3*x2^3*x3 - 2*x1^3*x2^3 - x1^3*x2^2*x3 + x1^3*x2^2 - 2*x1^2*x2^4*x3^2 + 3*x1^2*x2^3*x3^2 - x1^2*x2^3*x3 + 3*x1^2*x2^2*x3 - 2*x1^2*x2*x3^2 - 2*x1^2*x2*x3 + x1^2*x3^2 + x1*x2^3*x3^3 + x1*x2^2*x3^2 - x1*x2*x3^3 - 2*x1*x2*x3^2 + x1*x3^2 - x2*x3^4 + x3^4 - x3^3, x3 - 1, x1^2*x2^3 - x1^2*x2^2 - x1*x2^2*x3 + x3^2 - x3, x1^3*x2^3 - 2*x1^3*x2^2 + x1^3*x2 + x1^2*x2^3*x3 - x1^2*x2^2*x3 + x1^2*x2^2 - x1^2*x2*x3 - x1^2*x2 + x1^2*x3 - x1*x2^2*x3^2 + x1*x3 + x3^3 - x3^2, x1*x2^2 - x1*x2 - x3 + 1, x3, x1^2*x2^3 - 2*x1^2*x2^2 + x1^2*x2 + x1*x2^2*x3 + x1*x2^2 - 2*x1*x2*x3 - x1*x2 + x1*x3 - x2*x3 - x3^2 + 2*x3, x1^2*x2^4*x3 - x1^2*x2^3*x3 - x1^2*x2^3 - x1^2*x2^2*x3 + 2*x1^2*x2^2 + x1^2*x2*x3 - x1^2*x2 - x1*x2^3*x3 - x1*x2^2*x3^2 + 3*x1*x2^2*x3 - x1*x2^2 - x1*x2*x3 + x1*x2 + x1*x3^2 - x1*x3 + x3^2 - x3, x1^2*x2^3 - 2*x1^2*x2^2 + x1^2*x2 + x1*x2^3*x3 - x1*x2^2*x3 + x1*x2^2 - x1*x2*x3 - x1*x2 + x1*x3 - x2*x3^2 + x3, x2, 2*x1*x2 - 2*x1 - x3, x1^2*x2^3 - 2*x1^2*x2^2 + x1^2*x2 + x1*x2^2 - x1*x2*x3 - x1*x2 + x1*x3 + x3, x2 - 1, x1*x2^2 - x1*x2 - x2 - x3 + 1, x1^4*x2^7 - 4*x1^4*x2^6 + 6*x1^4*x2^5 - 4*x1^4*x2^4 + x1^4*x2^3 - x1^3*x2^6*x3 + x1^3*x2^6 + 2*x1^3*x2^5*x3 - 2*x1^3*x2^5 - 2*x1^3*x2^3*x3 + 2*x1^3*x2^3 + x1^3*x2^2*x3 - x1^3*x2^2 + x1^2*x2^5*x3 + 2*x1^2*x2^4*x3^2 - 2*x1^2*x2^4*x3 + x1^2*x2^4 - 5*x1^2*x2^3*x3^2 + x1^2*x2^3*x3 - 2*x1^2*x2^3 + 4*x1^2*x2^2*x3^2 + x1^2*x2^2 - x1^2*x2*x3^2 - x1*x2^4*x3^2 - x1*x2^3*x3^2 - x1*x2^3*x3 - x1*x2^2*x3^3 + 2*x1*x2^2*x3^2 + 3*x1*x2^2*x3 + 2*x1*x2*x3^3 - x1*x2*x3^2 - 2*x1*x2*x3 - x1*x3^3 + x1*x3^2 + x2^2*x3^3 + x2*x3^3 - x2*x3^2 - x3^3 + x3^2, x1^3*x2^5 - 3*x1^3*x2^4 + 3*x1^3*x2^3 - x1^3*x2^2 - x1^2*x2^4*x3 + x1^2*x2^4 + 2*x1^2*x2^3*x3 - 2*x1^2*x2^3 - x1^2*x2^2*x3 + x1^2*x2^2 + x1*x2^2*x3^2 + 2*x1*x2^2*x3 - 2*x1*x2*x3^2 - 2*x1*x2*x3 + x1*x3^2 - 2*x2*x3^2 + x3^2, x1*x2^2 - x1*x2 - x2*x3 + x3 - 1, x1^3*x2^4 - 2*x1^3*x2^3 + x1^3*x2^2 + x1^2*x2^3 - x1^2*x2^2 - x1*x2^2*x3^2 + x1*x2*x3^2 + x1*x2*x3 - x1*x3^2 + x3^3 - 2*x3^2, x1^3*x2^6*x3 - 2*x1^3*x2^5*x3 - x1^3*x2^5 + 3*x1^3*x2^4 + 2*x1^3*x2^3*x3 - 3*x1^3*x2^3 - x1^3*x2^2*x3 + x1^3*x2^2 - x1^2*x2^5*x3^2 - x1^2*x2^5*x3 + x1^2*x2^4*x3^2 + 3*x1^2*x2^4*x3 - x1^2*x2^4 + x1^2*x2^3*x3^2 - 3*x1^2*x2^3*x3 + 2*x1^2*x2^3 - x1^2*x2^2*x3^2 + x1^2*x2^2*x3 - x1^2*x2^2 + x1*x2^4*x3^2 + x1*x2^3*x3^3 - x1*x2^3*x3^2 + x1*x2^3*x3 - x1*x2^2*x3^3 + 2*x1*x2^2*x3^2 - 3*x1*x2^2*x3 - x1*x2*x3^3 - x1*x2*x3^2 + 2*x1*x2*x3 + x1*x3^3 - x1*x3^2 - x2^2*x3^3 - x2*x3^3 + x2*x3^2 + x3^3 - x3^2, x1*x2 - x1 - 1, x1^3*x2^3 - 2*x1^3*x2^2 + x1^3*x2 + x1^2*x2^2 - x1^2*x2*x3 - x1^2*x2 + x1^2*x3 + x1*x3 - x3^2, x1^2*x2^4 - x1^2*x2^3 - x1^2*x2^2 + x1^2*x2 - x1*x2^2*x3 + x1*x2^2 - x1*x2 + x1*x3 - x2^2*x3 + x2*x3 + x3, x1^2*x2^3 - 2*x1^2*x2^2 + x1^2*x2 + x1*x2^3*x3 - x1*x2^2*x3 + x1*x2^2 - x1*x2*x3 - x1*x2 + x1*x3 - x2^2*x3 - x2*x3^2 + x2*x3 + x3, x1^2*x2^2 - x1^2*x2 - x1*x3 - x3, x1^4*x2^5 - 3*x1^4*x2^4 + 3*x1^4*x2^3 - x1^4*x2^2 - x1^3*x2^4*x3 + x1^3*x2^4 + 2*x1^3*x2^3*x3 - 2*x1^3*x2^3 - x1^3*x2^2*x3 + x1^3*x2^2 - x1^2*x2^3*x3 + x1^2*x2^2*x3^2 + 3*x1^2*x2^2*x3 - 2*x1^2*x2*x3^2 - 2*x1^2*x2*x3 + x1^2*x3^2 + 2*x1*x2^2*x3^2 - 3*x1*x2*x3^2 + x1*x3^2 - x2*x3^3, x1*x2 - 1, x1^4*x2^5 - 3*x1^4*x2^4 + 3*x1^4*x2^3 - x1^4*x2^2 + x1^3*x2^5*x3 - 3*x1^3*x2^4*x3 + x1^3*x2^4 + 3*x1^3*x2^3*x3 - 2*x1^3*x2^3 - x1^3*x2^2*x3 + x1^3*x2^2 - x1^2*x2^4*x3^2 - x1^2*x2^4*x3 + x1^2*x2^3*x3^2 + x1^2*x2^3*x3 + x1^2*x2^2*x3^2 + 2*x1^2*x2^2*x3 - 2*x1^2*x2*x3^2 - 2*x1^2*x2*x3 + x1^2*x3^2 + x1*x2^3*x3^2 + x1*x2^2*x3^3 - x1*x2*x3^3 - 2*x1*x2*x3^2 + x1*x3^2 - x2*x3^3, x1^4*x2^5 - 3*x1^4*x2^4 + 3*x1^4*x2^3 - x1^4*x2^2 - x1^3*x2^4*x3 + x1^3*x2^4 + 2*x1^3*x2^3*x3 - 2*x1^3*x2^3 - x1^3*x2^2*x3 + x1^3*x2^2 - x1^2*x2^3*x3 + x1^2*x2^2*x3^2 + 3*x1^2*x2^2*x3 - 2*x1^2*x2*x3^2 - 2*x1^2*x2*x3 + x1^2*x3^2 + x1*x2^2*x3^2 - 2*x1*x2*x3^2 + x1*x3^2 - x3^3, x1^4*x2^5 - 3*x1^4*x2^4 + 3*x1^4*x2^3 - x1^4*x2^2 + x1^3*x2^4 - 2*x1^3*x2^3 + x1^3*x2^2 - x1^2*x2^3*x3^2 - x1^2*x2^3*x3 + 2*x1^2*x2^2*x3^2 + 3*x1^2*x2^2*x3 - 2*x1^2*x2*x3^2 - 2*x1^2*x2*x3 + x1^2*x3^2 + x1*x2^2*x3^2 + x1*x2*x3^3 - 2*x1*x2*x3^2 - x1*x3^3 + x1*x3^2 - x3^3, x1^4*x2^6 - 2*x1^4*x2^5 + 2*x1^4*x2^3 - x1^4*x2^2 - x1^3*x2^5*x3 + x1^3*x2^4*x3 + x1^3*x2^4 + x1^3*x2^3*x3 - 2*x1^3*x2^3 - x1^3*x2^2*x3 + x1^3*x2^2 - x1^2*x2^4*x3 + x1^2*x2^3*x3^2 + x1^2*x2^3*x3 - x1^2*x2^2*x3^2 + 2*x1^2*x2^2*x3 - x1^2*x2*x3^2 - 2*x1^2*x2*x3 + x1^2*x3^2 + x1*x2^3*x3^2 - 2*x1*x2*x3^2 + x1*x3^2 - x2*x3^3, x1^2*x2^2 - x1^2*x2 - x3, x1^2*x2^3 - 2*x1^2*x2^2 + x1^2*x2 + x1*x2^2*x3 + x1*x2^2 - 2*x1*x2*x3 - x1*x2 + x1*x3 - x3^2 + x3, x1^2*x2^4 - x1^2*x2^3 - x1^2*x2^2 + x1^2*x2 - x1*x2^2*x3 + x1*x2^2 - x1*x2 + x1*x3 + x3, x1^2*x2^3 - 2*x1^2*x2^2 + x1^2*x2 + x1*x2^2 - x1*x2*x3 - x1*x2 + x1*x3 - x2*x3^2 + x3, x1^2*x2^3 - 2*x1^2*x2^2 + x1^2*x2 + x1*x2^2*x3 + x1*x2^2 - 2*x1*x2*x3 - x1*x2 + x1*x3 - x2*x3^2 + x3, x1^2*x2^3 - 2*x1^2*x2^2 + x1^2*x2 - x1*x2^2*x3 + x1*x2^2 - x1*x2 + x1*x3 + x3, x1*x2^2 - x1*x2 - x3, x1^4*x2^6 - 4*x1^4*x2^5 + 6*x1^4*x2^4 - 4*x1^4*x2^3 + x1^4*x2^2 - x1^3*x2^5*x3 + x1^3*x2^5 + 3*x1^3*x2^4*x3 - 3*x1^3*x2^4 - 3*x1^3*x2^3*x3 + 3*x1^3*x2^3 + x1^3*x2^2*x3 - x1^3*x2^2 + 2*x1^2*x2^4*x3^2 - x1^2*x2^4*x3 - 3*x1^2*x2^3*x3^2 + 4*x1^2*x2^3*x3 - x1^2*x2^2*x3^2 - 5*x1^2*x2^2*x3 + 3*x1^2*x2*x3^2 + 2*x1^2*x2*x3 - x1^2*x3^2 - x1*x2^3*x3^3 - x1*x2^2*x3^3 - x1*x2^2*x3^2 + 2*x1*x2*x3^3 + 2*x1*x2*x3^2 - x1*x3^2 + x2*x3^4, x1^3*x2^5 - 3*x1^3*x2^4 + 3*x1^3*x2^3 - x1^3*x2^2 + x1^2*x2^4*x3 + x1^2*x2^4 - 2*x1^2*x2^3*x3 - 2*x1^2*x2^3 + x1^2*x2^2*x3 + x1^2*x2^2 - x1*x2^3*x3^2 - x1*x2^3*x3 + 3*x1*x2^2*x3 - 2*x1*x2*x3 + x1*x3^2 + x2*x3^3 - x2*x3^2 + x3^2, x1^3*x2^5 - 3*x1^3*x2^4 + 3*x1^3*x2^3 - x1^3*x2^2 - x1^2*x2^4*x3 + x1^2*x2^4 + 2*x1^2*x2^3*x3 - 2*x1^2*x2^3 - x1^2*x2^2*x3 + x1^2*x2^2 + x1*x2^3*x3^2 - x1*x2^3*x3 + 3*x1*x2^2*x3 - 2*x1*x2*x3^2 - 2*x1*x2*x3 + x1*x3^2 - x2*x3^3 - x2*x3^2 + x3^2]