The realization space is
  [1   0   1    x1 - 1   0   1    0    1   -x1^3 + x1^2 - x1 + 1    -x1^2 - 1      x1 - 1]
  [0   1   1   -x1 - 1   0   0    1    1    x1^3 + x1^2 + x1 + 1      -x1 - 1   -x1^2 - 1]
  [0   0   0         0   1   1   -1   x1              -x1^3 + x1   -x1^2 - x1   x1^2 - x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (4*x1^12 - 12*x1^10 + 12*x1^8 - 4*x1^6)
avoiding the zero loci of the polynomials
RingElem[x1, x1 + 1, 2, x1^2 + 3, x1 - 1, x1^2 + 1, x1^2 + x1 + 2, x1^2 - x1 + 2]