While I was waiting on the above info, I looked at the details you posted and noticed some of the teeth (or zig zag detail) appear to bend inward; perhaps this was done to hold the bands in place. The third image shows the teeth or zig zag detail bent a little toward the coral. One could tap all of the ends of the teeth on the loose bands ever so lightly with a small hammer (with the head covered with a thin fabric) to bend them slightly inward toward the coral. In doing so any concern about adhesives on the porous coral and the paper loosening and falling out is eliminated.

I read Jake's post about pitch and carbon being the original adhesive placed under the bands before they were brazed in place. How does one get it under silver and gold bands that are already in place? Is it thinned so it could be applied as a dilute liquid as described above?

http://www.wumag.kiev.ua/wumag_old/archiv/1_97/dress.htm

From the Soft Flex site: Harvests of red corals from Mediterranean waters have been reduced by 66 percent from 1985 to 2001. So much so, that the jewelry making industry there, (once famous for its red corals) is forced to import.
A record of combined harvest from the Mediterranean Sea and the Western Pacific Ocean shows that 445 metric tons were harvested in 1984 compared to just 50 metric tons in 2004! It has taken only 20 years to deplete the harvest to such extreme lows. (A little trivia here: 1 metric ton=2,205 pounds!)

I suggest that everyone consider not buying any contemporary coral!