In modern tape storage, user data are recorded and retrieved along multiple tracks of rapidly moving, flexible magnetic medium that give rise to a variety of channel impediments including occasional long erasures, more frequent amplitude fades as well as a large amount of random errors. This work considers reliable recovery of data from such tape channels using a novel concatenation of an inner Reed-Solomon (RS) code and an outer nonbinary low-density parity-check (LDPC) code. This particular concatenation scheme and a highly tailored iterative decoding algorithm are chosen to efficiently handle the assortment of the tape channel impediments while meeting the stringent target error rate constraint as well as key practical requirements of the mass tape storage system. Despite the use of a nonbinary LDPC code, the proposed scheme allows excellent performance-complexity tradeoffs. In stark contrast to any existing coding schemes that involve LDPC codes, the proposed concatenation strategy allows semianalytic error rate performance evaluation at rates below what is possible using modern computers, thus providing an ability to ensure satisfactory low-error-rate performance.