The realization space is
  [1   1   0   0                               x1 - 1                              x1 - 1   0                                                           x1 - 1                                                           x1 - 1                               x1 - 1    1]
  [0   1   1   1   x1^2*x2 - x1^2 + 4*x1*x2 - x1 - x2   x1^2*x2 - x1^2 + 4*x1*x2 - x2 - 1   0                                                                0                                                           x1 - 1   x1^2*x2 - x1^2 + 4*x1*x2 - x1 - x2   x1]
  [0   1   1   0                                    0                              x1 - 1   1   x1^2*x2^2 - 2*x1^2*x2 + x1^2 + 4*x1*x2^2 - 4*x1*x2 + x1 - x2^2   x1^2*x2^2 - 2*x1^2*x2 + x1^2 + 4*x1*x2^2 - 4*x1*x2 + x1 - x2^2                           x1*x2 - x2   x2]
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[x1^2*x2 - 2*x1^2 + 4*x1*x2 - x2, x2, x2 - 1, x1^2*x2 - x1^2 + 4*x1*x2 - x1 - x2, x1^3*x2^2 - 2*x1^3*x2 + x1^3 + 4*x1^2*x2^2 - 4*x1^2*x2 + x1^2 - x1*x2^2 - x1*x2 + x2, x1 - 1, x1^2*x2 - x1^2 + 4*x1*x2 - x2 - 1, x1^2*x2^2 - 2*x1^2*x2 + 2*x1^2 + 4*x1*x2^2 - 5*x1*x2 - x1 - x2^2 + x2 + 1, x1^4*x2^3 - 4*x1^4*x2^2 + 5*x1^4*x2 - 2*x1^4 + 8*x1^3*x2^3 - 21*x1^3*x2^2 + 14*x1^3*x2 - 2*x1^3 + 14*x1^2*x2^3 - 15*x1^2*x2^2 + 4*x1^2*x2 - 8*x1*x2^3 + 9*x1*x2^2 - 4*x1*x2 + x2^3 - x2^2 + x2, x1^4*x2^2 - 3*x1^4*x2 + 2*x1^4 + 8*x1^3*x2^2 - 12*x1^3*x2 + 14*x1^2*x2^2 + 2*x1^2*x2 + 3*x1^2 - 8*x1*x2^2 - 4*x1*x2 - 2*x1 + x2^2 + x2 + 1, x1, x1^2*x2^2 - 2*x1^2*x2 + x1^2 + 4*x1*x2^2 - 4*x1*x2 + x1 - x2^2, x1^2*x2 - x1^2 + 4*x1*x2 - x2, x1^2*x2^2 - 2*x1^2*x2 + 2*x1^2 + 4*x1*x2^2 - 5*x1*x2 - x2^2 + x2, x1^2*x2 - 2*x1^2 + 4*x1*x2 + x1 - x2 - 1, x1^2*x2^2 - x1^2*x2 - x1^2 + 4*x1*x2^2 + x1 - x2^2 - x2, x1^2*x2 - 2*x1^2 + 5*x1*x2 - 2*x2, x1^2*x2^2 - 2*x1^2*x2 + 2*x1^2 + 4*x1*x2^2 - 6*x1*x2 - x2^2 + 2*x2, x1 - x2, x1^2*x2 - x1^2 + 4*x1*x2 - 2*x1 - x2 + 1, x1^2*x2^2 - x1^2*x2 + 4*x1*x2^2 - x1*x2 - x1 - x2^2 + 1, x1*x2 - x1 + 3*x2 - 1, x1^4*x2^3 - 3*x1^4*x2^2 + 3*x1^4*x2 - x1^4 + 8*x1^3*x2^3 - 16*x1^3*x2^2 + 9*x1^3*x2 - x1^3 + 14*x1^2*x2^3 - 14*x1^2*x2^2 + 5*x1^2*x2 - 2*x1^2 - 8*x1*x2^3 + 3*x1*x2 + x1 + x2^3 + x2^2 - 1, x1^2*x2^2 - x1^2*x2 + 4*x1*x2^2 - x1 - x2^2 - x2 + 1, x1^4*x2^3 - 3*x1^4*x2^2 + 3*x1^4*x2 - x1^4 + 8*x1^3*x2^3 - 18*x1^3*x2^2 + 12*x1^3*x2 - 2*x1^3 + 14*x1^2*x2^3 - 20*x1^2*x2^2 + 6*x1^2*x2 - x1^2 - 8*x1*x2^3 + 10*x1*x2^2 + x2^3 - x2^2 - x2]