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