The realization space is
  [1   0   1   x2 + x3 - 2   0   1    0             1           x2 + x3 - 2                                                             x3 - 1    1]
  [0   1   1        x3 - 1   0   0    1             1                x3 - 1   x1*x3^2 - x1*x3 + x1 - x2*x3^2 + x2*x3 - x2 - x3^3 + 2*x3^2 - x3   x2]
  [0   0   0             0   1   1   -1   x2 + x3 - 1   x2*x3 + x3^2 - 2*x3                                                         x1*x3 - x1   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal with 4 generators
avoiding the zero loci of the polynomials
RingElem[x1 - x3, x2, x2 + x3 - 1, x2^2 + x2*x3 - 2*x2 + x3^2 - 2*x3 + 1, x2 - 1, x3, x3 - 1, x2 + x3, x2^2 + 2*x2*x3 - 2*x2 + x3^2 - 3*x3 + 1, x1*x2*x3 - x1 - x2*x3 - x3^2 + 2*x3, x1*x2^2 + x1*x2*x3 - 2*x1*x2 - x2^2 - x2*x3 + 2*x2 - x3^2 + 2*x3 - 1, x1 - 1, x1*x2 + x1*x3 - 2*x1 - x2 - x3^2 + x3 + 1, x1*x2^2 + x1*x2*x3 - x1*x2 - x1*x3 + x1 - x2^2 - x2*x3 + x2, x1 - x2 - x3 + 1, x1*x2 - x2 - x3 + 1, x1*x2 + x1*x3 - x1 - x2, x1*x2^2 + 2*x1*x2*x3 - 3*x1*x2 + x1*x3^2 - 3*x1*x3 + 2*x1 - x2^2 - x2*x3 + 2*x2 - x3^2 + 2*x3 - 1, x1, x2*x3 + x3^2 - x3 - 1, x2 + x3 - 2]