A core operation in data discovery is to find joinable tables for a given table. Real-world tables include both unary and n-ary join keys. However, existing table discovery systems are optimized for unary joins and are ineffective and slow in the existence of n-ary keys. In this paper, we introduce MATE, a table discovery system that leverages a novel hash-based index that enables n-ary join discovery through a space-efficient super key. We design a filtering layer that uses a novel hash, XASH. This hash function encodes the syntactic features of all column values and aggregates them into a super key, which allows the system to efficiently prune tables with non-joinable rows. Our join discovery system is able to prune up to 1000x more false positives and leads to over 60x faster table discovery in comparison to state-of-the-art.