Quantum communication networks have the potential to revolutionise information and communication technologies. A crucial yet challenging functionality required in any network is the ability to guarantee the anonymity of two parties, the Sender and the Receiver, when they wish to transmit a message through the network. Such anonymity is an increasingly valuable commodity in our information age. Here, we present a new protocol for players in a network to communicate both classical and quantum messages in a way that protects identity. Our work combines the power of classical and quantum protocols in a novel way, guaranteeing security against untrusted sources. As required for a realistic network, we ensure anonymity even in the presence of malicious parties. We define error-tolerant notions of anonymity, essential for realistic implementations, which we show can be achieved. Crucially, compared to previous results, we demonstrate a dramatic reduction in the required resources, leading to a practical protocol that can be performed with currently available experimental technology. (accepted by Physical Review Letters, arXiv:1811.04729)