Recently, a compressed index for similar strings, called the FM-index of alignment (FMA), has been proposed with the functionalities of pattern search and random access. The FMA is quite efficient in space requirement and pattern search time, but it is applicable only for an alignment of similar strings without gaps. In this paper we propose the FM-index of alignment with gaps, a realistic index for similar strings, which allows gaps in their alignment. For this, we design a new version of the suffix array of alignment by using alignment transformation and a new definition of the alignment-suffix. The new suffix array of alignment enables us to support the LF-mapping and backward search, the key functionalities of the FM-index, regardless of gap existence in the alignment. We experimentally compared our index with RLCSA due to Makinen et al. on 100 genome sequences from the 1000 Genomes Project. The index size of our index is less than one third of that of RLCSA.