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