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