The realization space is
  [1   0   1                          x3   0   1                                                                              0                                                                         x3^2    1      x3    1]
  [0   1   1   -x1*x2*x3 + x1*x2 + x1*x3   0   0                                                                           x3^2                                                                         x3^2   x2   x1*x2   x2]
  [0   0   0                           0   1   1   5*x1*x2*x3 - 2*x1*x2 + x1*x3^2 - 2*x1*x3 - 2*x2*x3^2 + x2*x3 - 2*x3^2 + 2*x3   5*x1*x2*x3 - 2*x1*x2 + x1*x3^2 - 2*x1*x3 - 2*x2*x3^2 + x2*x3 - x3^2 + 2*x3    1   x1*x3   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal with 7 generators
avoiding the zero loci of the polynomials
RingElem[x1 - x3, x2*x3 - x2 - x3, x1, x2, 5*x1*x2^2*x3 - 2*x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - 2*x2^2*x3^2 + x2^2*x3 - 2*x2*x3^2 + 2*x2*x3 - x3^3, 5*x1*x2^2*x3 - 2*x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - 2*x2^2*x3^2 + x2^2*x3 - x2*x3^2 + 2*x2*x3 - x3^3, x3 - 1, x2 - 1, x1*x2*x3 - x1*x2 - x1*x3 + x2*x3, 5*x1*x2*x3 - 2*x1*x2 + x1*x3^2 - 2*x1*x3 - 2*x2*x3^2 + x2*x3 - x3^3 - x3^2 + 2*x3, 5*x1*x2*x3 - 2*x1*x2 + x1*x3^2 - 2*x1*x3 - 2*x2*x3^2 + x2*x3 - x3^2 + 2*x3, 5*x1*x2^2*x3 - 2*x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - 2*x2^2*x3^2 + x2^2*x3 - 2*x2*x3^2 + 2*x2*x3 - x3^3 + x3^2, 5*x1*x2*x3 - 2*x1*x2 + x1*x3^2 - 2*x1*x3 - 2*x2*x3^2 + x2*x3 - 2*x3^2 + 2*x3, 5*x1*x2^2*x3 - 2*x1*x2^2 + x1*x2*x3^2 - 7*x1*x2*x3 + 2*x1*x2 - x1*x3^2 + 2*x1*x3 - 2*x2^2*x3^2 + x2^2*x3 + x2*x3 - x3^3 + 2*x3^2 - 2*x3, 5*x1^2*x2^2*x3^2 - 7*x1^2*x2^2*x3 + 2*x1^2*x2^2 + x1^2*x2*x3^3 - 8*x1^2*x2*x3^2 + 4*x1^2*x2*x3 - x1^2*x3^3 + 2*x1^2*x3^2 - 2*x1*x2^2*x3^3 + 8*x1*x2^2*x3^2 - 3*x1*x2^2*x3 + x1*x2*x3^3 + x1*x2*x3^2 - 2*x1*x2*x3 + 2*x1*x3^3 - 2*x1*x3^2 - 2*x2^2*x3^3 + x2^2*x3^2 - 2*x2*x3^3 + 2*x2*x3^2 - x3^4, x2 + x3 - 1, x1*x2*x3^2 - 2*x1*x2*x3 + x1*x2 - x1*x3^2 + x1*x3 - x2*x3, x3, x1*x2*x3 - x1*x2 - x1*x3 + x3, x1 - 1, 5*x1^2*x2^2*x3 - 2*x1^2*x2^2 + x1^2*x2*x3^2 - 2*x1^2*x2*x3 - 7*x1*x2^2*x3^2 + 3*x1*x2^2*x3 - x1*x2*x3^3 + 2*x1*x2*x3 - x1*x3^3 + 2*x2^2*x3^3 - x2^2*x3^2 + 2*x2*x3^3 - 2*x2*x3^2 + x3^3, 5*x1*x2*x3 - 2*x1*x2 - 2*x1*x3 - 2*x2*x3^2 + x2*x3 - x3^2 + 2*x3, x1*x2 - x3, 5*x1*x2^2*x3 - 2*x1*x2^2 - 6*x1*x2*x3 + 2*x1*x2 + 2*x1*x3 - 2*x2^2*x3^2 + x2^2*x3 + x2*x3^2 + x2*x3 - 2*x3, 5*x1^2*x2^2*x3 - 2*x1^2*x2^2 + x1^2*x2*x3^2 - 2*x1^2*x2*x3 - 2*x1*x2^2*x3^2 + x1*x2^2*x3 - 2*x1*x2*x3^2 + 2*x1*x2*x3 - x1*x3^3 + x3^3, 5*x1^2*x2^2*x3 - 2*x1^2*x2^2 + x1^2*x2*x3^2 - 2*x1^2*x2*x3 - 2*x1*x2^2*x3^2 + x1*x2^2*x3 - 7*x1*x2*x3^2 + 4*x1*x2*x3 - 2*x1*x3^3 + 2*x1*x3^2 + 2*x2*x3^3 - x2*x3^2 + 2*x3^3 - 2*x3^2, x1*x2 + x1*x3 - x3, x1*x2*x3 - x1*x2 - x1*x3 - x2*x3 + x3, 5*x1*x2^2*x3 - 2*x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - 2*x2^2*x3^2 + x2^2*x3 - x2*x3^2 + 2*x2*x3 - x3^2, 5*x1^2*x2^2*x3^2 - 7*x1^2*x2^2*x3 + 2*x1^2*x2^2 + x1^2*x2*x3^3 - 8*x1^2*x2*x3^2 + 4*x1^2*x2*x3 - x1^2*x3^3 + 2*x1^2*x3^2 - 2*x1*x2^2*x3^3 + 8*x1*x2^2*x3^2 - 3*x1*x2^2*x3 + x1*x2*x3^3 + x1*x2*x3^2 - 2*x1*x2*x3 + 2*x1*x3^3 - 2*x1*x3^2 - 2*x2^2*x3^3 + x2^2*x3^2 - x2*x3^3 + 2*x2*x3^2 - x3^3, 5*x1*x2^2*x3 - 2*x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - 2*x2^2*x3^2 + x2^2*x3 - 2*x2*x3^2 + 2*x2*x3 - x3^2, 5*x1^2*x2^2*x3^2 - 7*x1^2*x2^2*x3 + 2*x1^2*x2^2 + x1^2*x2*x3^3 - 8*x1^2*x2*x3^2 + 4*x1^2*x2*x3 - x1^2*x3^3 + 2*x1^2*x3^2 - 2*x1*x2^2*x3^3 + 8*x1*x2^2*x3^2 - 3*x1*x2^2*x3 + x1*x2*x3^3 + x1*x2*x3^2 - 2*x1*x2*x3 + 2*x1*x3^3 - 2*x1*x3^2 - 2*x2^2*x3^3 + x2^2*x3^2 - 2*x2*x3^3 + 2*x2*x3^2 - x3^3, 5*x1^2*x2^2*x3^2 - 7*x1^2*x2^2*x3 + 2*x1^2*x2^2 + x1^2*x2*x3^3 - 8*x1^2*x2*x3^2 + 4*x1^2*x2*x3 - x1^2*x3^3 + 2*x1^2*x3^2 - 2*x1*x2^2*x3^3 + 3*x1*x2^2*x3^2 - x1*x2^2*x3 + 3*x1*x2*x3^2 - 2*x1*x2*x3 + 2*x1*x3^3 - 2*x1*x3^2 - x3^3]