In cryptographic protocols, in particular RFID protocols, exclusive-or (XOR) operations are common. Due to the inherent complexity of faithful models of XOR, there is only limited tool support for the verification of cryptographic protocols using XOR. In this paper, we improve the TAMARIN prover and its underlying theory to deal with an equational theory modeling XOR operations. The XOR theory can be combined with all equational theories previously supported, including user-defined equational theories. This makes TAMARIN the first verification tool for cryptographic protocols in the symbolic model to support simultaneously this large set of equational theories, protocols with global mutable state, an unbounded number of sessions, and complex security properties including observational equivalence. We demonstrate the effectiveness of our approach by analyzing several protocols that rely on XOR, in particular multiple RFID-protocols, where we can identify attacks as well as provide proofs.