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