Having just got this achievement I thought I would post up a solution with screenshots to show exactly what needs to be done.
There are nine main assassinations that you need to carry out. One in each of the Rich, Middle and Poor districts of Damascus, Acre and Jerusalem. You MUST NOT perform the main assassination in a district until you have completed all the side mission within each one - saving citizens, pick-pocketing, eavesdropping and informer missions - as well as synchronizing all the high points (which you should do anyway to make it easier to locate all the side missions). Only then should you assassinate the main target. Do this for every district as you play through and you will be fine. This achievement has NOTHING to do with your synchronisation health bar that is displayed in your HUD and you do not need to collect all the flags (except for the ones featured in the Informer side missions, obviously!)
Here's what you should see when you have completed everything you need to and before you have visited the Assassins Bureau prior to the main assassination to collect your feather. The left set of ten strands is a sequence/assassination already completed and the right set of strands is the current sequence/assassination. You can see there are two black unfilled strands.
Once you have got to the stage shown above, visiting the Assassins Bureau and collecting your feather will fill one of the two remaining strands. Performing the assassination will fill the last strand as shown here:
For the very last assassination in the game there will be an extra black strand remaining at this point. Don't worry as it's story related. Provided you have done them all up to this point in the game the achievement will pop for you once the final target is killed. I did not perform all the assassinations stealthily, in fact most of them were pretty messy fights. What you need to do to get this achievement is exactly what I have described and nothing more.
This video shows what you should see when you have completed everything you need to and once you have collected your feather from the Assassins Bureau but before carrying out the main assassination. The filled strands are the side missions already complete and performing the main assassination will fill the last strand.