What I think you are seeing is one of the rare cases where balancing seriously degrades the accuracy of the computed eigenvectors. For a reference, see: David S. Watkins, A case where balancing is harmful, Electron. Trans. Numer. Anal., 23 (2006), pp. 1-4. which is available on his website at http://www.math.wsu.edu/faculty/watkins/pdfiles/balbad.pdf
. Watkins writes in his conclusion:
"Balancing sometimes seriously degrades accuracy. In particular, one should not balance a matrix after it has been transformed to Hessenberg form. However, we must emphasize that balancing is usually not harmful and often very beneficial. When in doubt, balance."
The solution is as you discovered to use the 'nobalance' option in Matlab. In LAPACK, you can use dgeevx instead of dgeev to optionally turn off balancing. I tried your matrix using a modified version of dgeev with balancing turned off and it worked fine.