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