The realization space is
  [1                   x3       x2 - x3   1   0              x2 - x3   0   x2                       x2*x3    0    1]
  [1   -x1*x2 - x2 + 3*x3             0   0   1   -x1*x2 - x2 + 2*x3   0   x2   -x1*x2^2 - x2^2 + 3*x2*x3    1   x2]
  [0                   x3   -x1*x2 + x3   0   0          -x1*x2 + x3   1   x3                        x3^2   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*x2^2 - 3*x1*x2*x3 + x3^2)
avoiding the zero loci of the polynomials
RingElem[x1*x2 - x1 - x3, x1^2*x2 + x1*x2*x3 + x1*x2 - 3*x1*x3 - x3^2 + x3, x2 - x3 - 1, x1*x2 - x3, x1, x1^2*x2 + x1*x2^2 - x1*x2*x3 - 2*x1*x3 - x2*x3 + x3^2 + x3, x2 - 1, x1^2*x2^2 + x1*x2^2*x3 + x1*x2^2 - 3*x1*x2*x3 - x2*x3^2 + x3^2, x1*x2^2 + x2^2 - x2*x3 - x3, x2 - x3, x1*x2 + x2*x3 + x2 - 3*x3, x1^2*x2^2 + 2*x1*x2^2*x3 + x1*x2^2 - x1*x2*x3^2 - 4*x1*x2*x3 + x2^2*x3 - 4*x2*x3^2 - x2*x3 + 2*x3^3 + 3*x3^2, x1*x2 + x2 - 2*x3, x3, x1^2*x2^3 + 2*x1*x2^3*x3 + x1*x2^3 - 2*x1*x2^2*x3^2 - 4*x1*x2^2*x3 + x1*x2*x3^2 + x2^3*x3 - 5*x2^2*x3^2 - x2^2*x3 + 4*x2*x3^3 + 4*x2*x3^2 - 2*x3^3, x1*x2 + 2*x2 - 3*x3, x1*x2 - 2*x3, x2, x1*x2 + x2^2 - x2*x3 + x2 - 2*x3, x1*x2^2 - x1*x2*x3 - x1*x2 - 3*x2*x3 + 3*x3^2 + x3, x1^2*x2^2 + 2*x1*x2^2*x3 + x1*x2^2 - 2*x1*x2*x3^2 - 3*x1*x2*x3 + 2*x2^2*x3 - 7*x2*x3^2 + 5*x3^3 + x3^2, x1*x2 + x2*x3 - x3^2 - x3, x3 - 1, x1*x2^2 - x1*x2 - 2*x2*x3 + x3^2 + x3, x1^2*x2^2 + 2*x1*x2^2*x3 + x1*x2^2 - x1*x2*x3^2 - 4*x1*x2*x3 + 2*x2^2*x3 - 6*x2*x3^2 - x2*x3 + 3*x3^3 + 3*x3^2, x1*x2 + 2*x2 - 2*x3 - 1, x1^2*x2^2 + x1*x2^2 - 2*x1*x2*x3 + x3^2, x1^2*x2^3 - x1^2*x2^2*x3 + x1*x2^3 - 3*x1*x2^2*x3 + 3*x1*x2*x3^2 - x3^3, x1^2*x2^3 - 2*x1^2*x2^2*x3 + x1*x2^3 - 4*x1*x2^2*x3 + 5*x1*x2*x3^2 - x3^3, x1^2*x2^2 + x1*x2^2 - 2*x1*x2*x3 + x2*x3 - x3^2, x1^2*x2 + x1*x2 - 3*x1*x3 + x3, x1^2*x2^2 - 2*x1^2*x2*x3 + x1*x2^2 - 4*x1*x2*x3 + 5*x1*x3^2 + x2*x3 - 2*x3^2, x1^2*x2 - 2*x1*x3 + x3, x1 - 1, x1^2*x2^2 - x1^2*x2*x3 + x1*x2^2 - 3*x1*x2*x3 + 3*x1*x3^2 + x2*x3 - 2*x3^2, x1^2*x2 + x1*x2 - 2*x1*x3 + x3, x1*x2^2 - x3^2, x1*x2 + x2 - 3*x3, x1*x2^2 - 2*x1*x2*x3 + x2^2 - 5*x2*x3 + 5*x3^2, x1^2*x2^3 + x1*x2^3 - 3*x1*x2^2*x3 + x1*x2*x3^2 - x2^2*x3 + 4*x2*x3^2 - 3*x3^3, x1^2*x2^3 + x1*x2^3 - 4*x1*x2^2*x3 + x1*x2*x3^2 - x2^2*x3 + 4*x2*x3^2 - 2*x3^3, x1^2*x2^3 + x1*x2^3 - 3*x1*x2^2*x3 - x2^2*x3 + 3*x2*x3^2 - x3^3, x1^2*x2^3 + x1*x2^3 - x1*x2^2*x3 - 2*x1*x2*x3^2 + 2*x2^2*x3 - 6*x2*x3^2 + 5*x3^3, x1^2*x2^3 + x1*x2^3 - 2*x1*x2^2*x3 - x1*x2*x3^2 + x2^2*x3 - 3*x2*x3^2 + 3*x3^3]