The realization space is
  [1   1   x1 + 1   0   1   0   x1 + 1   x1 + 1   x1^2 + 2*x1 + 1   x1 + 1    0]
  [0   1        1   1   0   0        0        1     x1^2 + x1 + 1   x1 + 1    1]
  [0   0        0   0   1   1    -x1^2       x1      -x1^3 - x1^2       x1   x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (-x1^12 - 6*x1^11 - 12*x1^10 - 11*x1^9 - 5*x1^8 - x1^7)
avoiding the zero loci of the polynomials
RingElem[x1, x1 - 1, x1^2 + x1 + 1, 2*x1^2 + 2*x1 + 1, 2*x1 + 1, x1 + 1, x1^3 + 3*x1^2 + 2*x1 + 1, x1 + 2]