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