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