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