The realization space is
  [0   1   1   0   0   1                          x1                                                  x1                           1                          x1    1]
  [1   0   1   1   0   1   2*x1*x2 - x1*x3 + x1 - x2                  2*x1^2*x2 - x1^2*x3 + x1^2 - x1*x2   2*x1*x2 - x1*x3 + x1 - x2   2*x1*x2 - x1*x3 + x1 - x2   x2]
  [1   0   1   0   1   0                          x1   2*x1^2*x2 - x1^2*x3 + x1^2 - 3*x1*x2 + x1*x3 + x2                          x1                       x1*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 - x1*x3 + x1 - x2, x3, x3 - 1, 2*x1^2*x2 - x1^2*x3 + x1^2 - 3*x1*x2 + x2, x1 - x3, 2*x2 - x3, x1 - 1, 2*x1*x2*x3 - x1*x2 - x1*x3^2 + x1*x3 - x2*x3, 2*x1*x2*x3 - 3*x1*x2 - x1*x3^2 + 2*x1*x3 - x2*x3 + 2*x2 - x3, 2*x1*x2 - x1*x3 + x1 - 2*x2, 2*x1*x2*x3 - x1*x2 - x1*x3^2 + x1*x3 + x1 - x2*x3 - x3, 2*x1*x2 - x1*x3 - x2, x2, 2*x1^2*x2^2 - 3*x1^2*x2*x3 + x1^2*x2 + x1^2*x3^2 - x1^2*x3 - 3*x1*x2^2 + 2*x1*x2*x3 + x1*x2 + x2^2 - x2, 2*x1^2*x2^2 - 3*x1^2*x2*x3 - x1^2*x2 + x1^2*x3^2 - x1^2 - 3*x1*x2^2 + 2*x1*x2*x3 + 3*x1*x2 + x2^2 - x2, 2*x1*x2 - x1*x3 + x1 - x2, 2*x1*x2*x3 - x1*x2 - x1*x3^2 + x1 - x2*x3, x2 - x3 - 1, x2 - 1, x2 - x3, 2*x1^2*x2 - x1^2*x3 - 3*x1*x2 + 2*x1*x3 - x1 + x2, 2*x1^2*x2*x3 - 4*x1^2*x2 - x1^2*x3^2 + 3*x1^2*x3 - x1^2 - x1*x2*x3 + 4*x1*x2 - 2*x1*x3 + x1 - x2, 2*x1*x2*x3 - 2*x1*x2 - x1*x3^2 + 2*x1*x3 - x2*x3 + x2 - x3, 2*x1^2*x2 - 2*x1^2*x3 + x1^2 - 3*x1*x2 + x1*x3 + x2, 4*x1^3*x2^2 - 6*x1^3*x2*x3 + 4*x1^3*x2 + 2*x1^3*x3^2 - 3*x1^3*x3 + x1^3 - 8*x1^2*x2^2 + 7*x1^2*x2*x3 - 4*x1^2*x2 - x1^2*x3^2 + 2*x1^2*x3 - x1^2 + 5*x1*x2^2 - 2*x1*x2*x3 + x1*x2 - x2^2, 2*x1*x2 - 2*x1*x3 - x2, 2*x1*x2 - 2*x1*x3 + x1 - x2, 2*x1*x2 - x1*x3 + x1 - x2 - 1, 2*x1*x2 - x1*x3 - x2 - 1, x1, 2*x1*x2 - x1*x3 - x1 - x2, 2*x1^2*x2 - x1^2*x3 + x1^2 - 3*x1*x2 + x1*x3 + x2]