I'm sure black coral was originally widely used in Yemen as prayer beads, misbaha. Modern misbaha beads tend to be smaller than the black coral beads in your necklace though black coral prayer beads from the beginning of the 20thC are bigger (like yours) according to the photos from "Coral in the Traditional Ornaments of the Yemen".

The silver beads are interesting and I also have some which came from Yemen or possibly "via Yemen". I think that they may actually be modern beads made in Yemen. The interest in and prices paid for old, high quality silver work from Yemen could have stimulated a cottage industry in reproductions/ fakes depending on how they are represented. If the skill still exists to produce them. I think it probably does. One doubt that I have regarding these beads is the silver content. I've never tested the silver but it seems to be different from the range of alloys used in old work.

Your beads are much less patinated than mine but I think they are the same.

I have seen other Yemeni items here on BCN that I think may also be modern, high quality, high price Yemeni reproductions.
Brian