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