The realization space is
  [1   1   x1 - 1   0   0   1    1     x1 - 1          0   x1^2 - 1    x1^2 - 1]
  [1   0       x1   1   0   1    0         x1       x1^2       x1^2        x1^2]
  [0   0        0   0   1   1   x1   x1^2 - 1   x1^2 - 1   x1^2 - 1   x1^3 - x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal with 1 generator
avoiding the zero loci of the polynomials
RingElem[x1 - 1, x1, x1^2 + x1 + 1, x1^2 + 1, x1^3 - x1^2 - 1, x1 + 1, x1^2 - x1 - 1, x1^2 + x1 - 1, x1^3 - x1 - 1, x1^3 + x1^2 - 1]