The realization space is
  [1   1              1   0   1   0               1                 1               1                 1    0]
  [0   1   -x1^2 - 2*x1   1   0   0               0      -x1^2 - 2*x1             -x1                 1    1]
  [0   0              0   0   1   1   x1^2 + x1 + 1   x1^2 + 2*x1 + 1   x1^2 + x1 + 1   x1^2 + 2*x1 + 1   x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (x1^3 + 3*x1^2 + 2*x1 + 1, x1^10 + 7*x1^9 + 20*x1^8 + 29*x1^7 + 20*x1^6 + 2*x1^5 - 5*x1^4 - 2*x1^3)
avoiding the zero loci of the polynomials
RingElem[x1^2 + x1 + 1, x1 + 1, x1 - 1, x1, 2*x1^2 + x1 + 1, 2*x1^2 + 2*x1 + 1, x1^3 - x1 - 1, 2*x1 + 1, x1 + 2, x1^2 + x1 - 1, x1^2 + 3*x1 + 1, x1^2 + 2*x1 - 1, x1^3 + 2*x1^2 + x1 + 1]