Reusing an FM-index

Intuitively, if two strings $S_1$ and $S_2$ are sufficiently similar and we already have an FM-index for $S_1$ then, by storing a little extra information, we should be able to reuse parts of that index in an FM-index for $S_2$. We formalize this intuition and show that it can lead to significant space savings in practice, as well as to some interesting theoretical problems.