The Tor anonymity network depends on volunteers to operate relays, and might offer higher bandwidth with lower response latencies if more users could be incentivized to contribute relay bandwidth. We introduce TEARS, a system rewarding useful service with traffic priority. TEARS audits relays and rewards them with anonymous coins called Shallots, proportionally to bandwidth contributed. Shallots may be redeemed anonymously for PriorityPasses, which in turn may be presented to relays to request traffic priority. The PriorityPass construction enables relays to prevent double spending locally without leaking information. Unlike previous incentive proposals, TEARS incorporates transparent and distributed banking using protocols from distributed digital cryptocurrency systems like Bitcoin. Shallots are publicly-verifiable, minimizing reliance on and trust in banking authorities, making them auditable while naturally distributing bank functionality and associated overhead. Further, these distributed banking protocols resist denial-of-service attacks and can recover from catastrophic failures. TEARS may either be deployed in the existing Tor network or operate alongside it.