The realization space is
  [1   1          x2 - x3   0    1   x3          x2 - x3    1   0      x3    0]
  [0   1   -x1*x3 + x2*x3   1   x1   x2          x2 - x3   x1   0      x2    1]
  [0   1   -x1*x3 + x2*x3   0    0   x3   -x1*x3 + x2*x3   x2   1   x2*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[x3 - 1, x1 - x2, x3, x1, x2 - 1, x1 - x3, x1*x3 - x2, x1*x3 - x2 - x3 + 1, x1*x2*x3 - x1*x3 - x2^2 - x2*x3^2 + 2*x2*x3, x2, x1*x3 - 2*x2 + 1, x1*x2*x3 - x1*x3^2 - x1*x3 - x2^2 + x2*x3 + x3^2, x1*x3 - x2*x3 + x2 - x3, x1*x2 - x1*x3 - x1 + x3, x1*x3 - x2^2 + x2*x3 + x2 - x3^2 - x3, x2 - x3, x1*x3^2 - x1*x3 + x2^2 - x2*x3^2 - x2 + x3, x1*x3 - x2 + 1, x1*x3 - x3 + 1, x1*x2 - x1 - x2*x3 + x2, x1*x3^2 + x2^2 - x2*x3^2 - x2*x3, x1*x3 + x2^2 - 2*x2*x3, x1^2*x3 + x1*x2^2 - 2*x1*x2*x3 - x1*x2 + x2*x3, x1*x2*x3 - x1*x3 - x2*x3 + x2, x1^2*x3 + x1*x2^2 - x1*x2*x3 - x1*x2 - x2^2*x3 + x2^2, x1 - 1, x1^2*x3^2 - x1*x2*x3^2 - x1*x3^2 - x2^3 + 3*x2^2*x3 - x2*x3^2 - x2*x3 + x3^2, x1*x3 - x2^2, x1^2*x3^2 - 2*x1*x2*x3^2 - x2^3 + x2^2*x3^2 + 2*x2^2*x3 - x2*x3^2, x1*x2 - x2^2 + x2 - x3, x1^2*x3 - x1*x2*x3 - x1*x3 + x2^2 - x2 + x3]