The realization space is
  [1    x2   1   0       x3      x2    x2          x2*x3 + x3^2   0    0    1]
  [1   -x3   0   1   -x1*x2   x1*x2   -x3   -x1*x2^2 - x1*x2*x3   0    1    0]
  [1   -x3   0   0       x3     -x3     0       -x1*x2^2 + x3^2   1   x2   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal with 3 generators
avoiding the zero loci of the polynomials
RingElem[x2 + x3 - 1, x1*x2^4 + 2*x1*x2^3*x3 - 2*x1*x2^3 + x1*x2^2*x3^2 - 2*x1*x2^2*x3 + x1*x2^2 - x1*x2*x3^2 + 2*x2*x3^2 + x3^3 - x3^2, x2, x3, x1*x2^2 - x3^2 + x3, x1*x2^2 - x3^2, x1*x2^3 + 2*x1*x2^2*x3 - x1*x2^2 + x1*x2*x3^2 + x3^2, x1, x1*x2 + x3, x1^2*x2^4 + 2*x1^2*x2^3*x3 - x1^2*x2^3 + x1^2*x2^2*x3^2 - x1*x2^4*x3 - 3*x1*x2^3*x3^2 + x1*x2^3*x3 - 3*x1*x2^2*x3^3 + x1*x2^2*x3^2 - x1*x2^2*x3 - x1*x2*x3^4 + x1*x2*x3^2 - x2*x3^3 - x3^4 + x3^3, x1*x2^2 + x2*x3^2 + x3^3 - x3^2, x1^2*x2^5 + 2*x1^2*x2^4*x3 - x1^2*x2^4 + x1^2*x2^3*x3^2 + x1*x2^4*x3^2 + 3*x1*x2^3*x3^3 - x1*x2^3*x3^2 + 3*x1*x2^2*x3^4 - x1*x2^2*x3^3 + 2*x1*x2^2*x3^2 + x1*x2*x3^5 + x2*x3^4 + x3^5 - x3^4, x1*x2^3*x3 - x1*x2^3 + 2*x1*x2^2*x3^2 - 3*x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^3 - x1*x2*x3^2 - x2*x3^2 - x3^2, x1*x2^3*x3 - x1*x2^3 + 2*x1*x2^2*x3^2 - 2*x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^3 + x3^3 - x3^2, x1^2*x2^6 + 4*x1^2*x2^5*x3 - 2*x1^2*x2^5 + 6*x1^2*x2^4*x3^2 - 4*x1^2*x2^4*x3 + x1^2*x2^4 + 4*x1^2*x2^3*x3^3 - 2*x1^2*x2^3*x3^2 + x1^2*x2^2*x3^4 - x1*x2^4*x3^2 - 3*x1*x2^3*x3^3 + 3*x1*x2^3*x3^2 - 3*x1*x2^2*x3^4 + 6*x1*x2^2*x3^3 - 2*x1*x2^2*x3^2 - x1*x2*x3^5 + 4*x1*x2*x3^4 + x1*x3^5 - x2*x3^4 - x3^5 + x3^4, x1*x2^2 + x1*x2*x3 + x1*x3^2 - x3^2, x1*x2^3 + 2*x1*x2^2*x3 - x1*x2^2 + x1*x2*x3^2 - x1*x2*x3 - x1*x3^2 + x3^2, x1*x2^3 + 2*x1*x2^2*x3 - x1*x2^2 + x1*x2*x3^2 - x2*x3^2 - x3^3 + x3^2, x1*x2^4 + 2*x1*x2^3*x3 - x1*x2^3 + x1*x2^2*x3^2 - x2^2*x3^2 - 2*x2*x3^3 + 2*x2*x3^2 - x3^4 + x3^3, x3 - 1, x2 + x3, x1^2*x2^4 + 2*x1^2*x2^3*x3 - x1^2*x2^3 + x1^2*x2^2*x3^2 - x1*x2^3*x3^2 + x1*x2^3*x3 - 2*x1*x2^2*x3^3 + 2*x1*x2^2*x3^2 - x1*x2^2*x3 - x1*x2*x3^4 + x1*x2*x3^2 - x3^4 + x3^3, x1*x2^2 + 2*x1*x2*x3 - x1*x2 + x1*x3^2 - x3, x1*x2^3 + x1*x2^2*x3 - x1*x2^2 + x3^2, x1*x2^4 + 2*x1*x2^3*x3 + x1*x2^2*x3^2 + x1*x2^2*x3 - x1*x2^2 + x2*x3^2 + x3^2, x1*x2^4 + 2*x1*x2^3*x3 - x1*x2^3 + x1*x2^2*x3^2 - x2^2*x3 + x2*x3 + x3^2, x1*x2^4 + 2*x1*x2^3*x3 + x1*x2^2*x3^2 + x1*x2^2*x3 + x2*x3^2 + x2*x3 + x3^2, x1*x2^3 + 2*x1*x2^2*x3 - x1*x2^2 + x1*x2*x3^2 - x1*x2*x3 + x1*x2 - x1*x3^2 + x3^2 + x3, x1*x2^2 + x3, x2 - 1, x1*x2^3 + 2*x1*x2^2*x3 + x1*x2*x3^2 + x1*x2*x3 + x3^2, x1 - 1, x1^2*x2^6 + 4*x1^2*x2^5*x3 - 2*x1^2*x2^5 + 6*x1^2*x2^4*x3^2 - 4*x1^2*x2^4*x3 + x1^2*x2^4 + 4*x1^2*x2^3*x3^3 - 2*x1^2*x2^3*x3^2 + x1^2*x2^2*x3^4 - 2*x1*x2^4*x3 - 4*x1*x2^3*x3^2 + 2*x1*x2^3*x3 - 2*x1*x2^2*x3^3 + x1*x2^2*x3^2 + 2*x1*x2*x3^3 + x1*x3^4 - 2*x2*x3^3 - x3^4, x1^2*x2^6 + 4*x1^2*x2^5*x3 - x1^2*x2^5 + 6*x1^2*x2^4*x3^2 - x1^2*x2^4*x3 + 4*x1^2*x2^3*x3^3 + x1^2*x2^3*x3^2 - x1^2*x2^3*x3 + x1^2*x2^2*x3^4 + x1^2*x2^2*x3^3 - x1*x2^4*x3 - x1*x2^3*x3^2 + x1*x2^3*x3 + x1*x2^2*x3^3 + x1*x2^2*x3^2 + x1*x2*x3^4 + 3*x1*x2*x3^3 + x1*x3^4 - x2*x3^3]