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