The realization space is
  [1       x1   1   0   1   0          1        1          1          x1    0]
  [1   x1 + 1   0   1   1   0          1        0   2*x1 + 1      x1 + 1    1]
  [0        0   0   0   1   1   2*x1 + 1   x1 + 1   2*x1 + 1   x1^2 + x1   x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (4*x1^7 + 8*x1^6 + 4*x1^5 - 6*x1^4 - 8*x1^3 - 2*x1^2)
avoiding the zero loci of the polynomials
RingElem[x1, x1 + 1, x1 - 1, 2*x1 + 1, 2*x1^2 - 2*x1 - 1, x1^2 - x1 - 1, 2, 2*x1^3 + 2*x1^2 - 2*x1 - 1, 2*x1^2 - 1, 2*x1^2 + 4*x1 + 1]