The realization space is
  [1   1   0   2   0   1   1   0   -6   4   4]
  [0   1   1   1   0   0   1   1   -3   1   3]
  [0   0   0   0   1   1   1   3   -5   3   3]
in the integer ring
within the vanishing set of the ideal
28ZZ
avoiding the zero loci of the polynomials
RingElem[3, 5, 2, 7]