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