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