The realization space is
  [1   1   0   0   1   1          0                                        x1*x2 - x2^2                                        x1*x2 - x2^2                             x2                                  x1 - x2]
  [0   1   1   0   0   1          1   -x1^3 + 3*x1^2*x2 - x1^2 - 2*x1*x2^2 - x1*x2 + x1   -x1^3 + 3*x1^2*x2 - x1^2 - 2*x1*x2^2 - x1*x2 + x1   x1^2 - 2*x1*x2 + x1 + x2 - 1   -x1^2 + 3*x1*x2 - x1 - 2*x2^2 - x2 + 1]
  [0   0   0   1   1   1   -x1 + x2                                      x1*x2^2 - x2^3                                   x1^2*x2 - x1*x2^2                          x1*x2                             x1*x2 - x2^2]
in the multivariate polynomial ring in 2 variables over ZZ
within the vanishing set of the ideal
Ideal with 3 generators
avoiding the zero loci of the polynomials
RingElem[2*x1^3 - 6*x1^2*x2 + 2*x1^2 + 4*x1*x2^2 + x1*x2 - 2*x1 - x2^2 + x2, 2*x1^3 - 6*x1^2*x2 + 3*x1^2 + 4*x1*x2^2 - x1*x2 - 2*x1 + x2, x1 - x2, x1^3 - 2*x1^2*x2 + x1^2 - x1*x2^2 + x1*x2 - x1 + 2*x2^3, 2*x1^3*x2 - x1^3 - 6*x1^2*x2^2 + 4*x1^2*x2 - x1^2 + 4*x1*x2^3 + 2*x1*x2^2 - 3*x1*x2 + x1 - 3*x2^3 + x2^2, x1^3 - 2*x1^2*x2 + x1^2 - x1*x2^2 + 2*x1*x2 - x1 + 2*x2^3 - x2^2, x1^2 - 3*x1*x2 + x1 + 2*x2^2 + x2 - 1, x1^2 - 2*x1*x2 + x1 + x2^2 + x2 - 1, x1^2 - 3*x1*x2 + x1 + 2*x2^2 - 1, x2, x2 - 1, x1^2 - 3*x1*x2 + 2*x1 + 2*x2^2 - x2 - 1, x1^2 - 3*x1*x2 + x1 + 2*x2^2, x1^2 - 3*x1*x2 + 2*x1 + 2*x2^2 - x2, x1^2 - 3*x1*x2 + 2*x1 + 2*x2^2 - 1, x1^2 - 4*x1*x2 + 2*x1 + 3*x2^2 - 1, x1, x1 - 1, x1^4 - 2*x1^3*x2 + x1^3 - x1^2*x2^2 + 2*x1^2*x2 - x1^2 + 2*x1*x2^3 - x1*x2 - x2^3 + x2^2, x1^4 - 2*x1^3*x2 + x1^3 - x1^2*x2^2 + 3*x1^2*x2 - x1^2 + 2*x1*x2^3 - 2*x1*x2^2 - x1*x2 + x2^2, x1^4 - 2*x1^3*x2 - x1^3 - x1^2*x2^2 + 8*x1^2*x2 - 3*x1^2 + 2*x1*x2^3 - 4*x1*x2^2 - 2*x1*x2 + 2*x1 - x2^3 + 2*x2^2 - x2, x1^4 - 2*x1^3*x2 - x1^3 - x1^2*x2^2 + 9*x1^2*x2 - 3*x1^2 + 2*x1*x2^3 - 6*x1*x2^2 - 2*x1*x2 + 2*x1 + 2*x2^2 - x2, 2*x1^3 - 6*x1^2*x2 + 2*x1^2 + 4*x1*x2^2 + 2*x1*x2 - 2*x1 - 2*x2^2 + x2, x1^3 - 3*x1^2*x2 + x1^2 + 2*x1*x2^2 + x1*x2 - x1 - x2^2 + x2, x1^3 - 3*x1^2*x2 + x1^2 + 2*x1*x2^2 - x1 + x2, x1^2 - 3*x1*x2 + x1 + x2 - 1, x1^2 - 2*x1*x2 + x1 - 1, x1^2 - 2*x1*x2 + x1 + x2 - 1, x1^2 - x1*x2 + x1 - 1, x1^3 - 3*x1^2*x2 + x1^2 + 2*x1*x2^2 + 2*x1*x2 - x1 - x2^2, x1^3 - 4*x1^2*x2 + x1^2 + 3*x1*x2^2 + 2*x1*x2 - x1 - x2^2, x1^3 - 3*x1^2*x2 + x1^2 + 2*x1*x2^2 + 2*x1*x2 - x1 - 2*x2^2, x1^3 - 3*x1^2*x2 + x1^2 + 2*x1*x2^2 + 2*x1*x2 - x1 - 2*x2^2 + x2, x1^3 - 3*x1^2*x2 + x1^2 + 3*x1*x2^2 + x1*x2 - x1 - x2^3, x1^3 - 3*x1^2*x2 + x1^2 + x1*x2^2 + 2*x1*x2 - x1 + x2^3 - x2^2, x1 - x2 + 1, x1 - x2 - 1]