We propose Tesseract, a secure real-time cryptocurrency exchange service. Existing centralized exchange designs are vulnerable to theft of funds, while decentralized exchanges cannot offer real-time cross-chain trades. All currently deployed exchanges are also vulnerable to frontrunning attacks. Tesseract overcomes these flaws and achieves a best-of-both-worlds design by using a trusted execution environment. The task of committing the recent trade data to independent cryptocurrency systems presents an all-or-nothing fairness problem, to which we present ideal theoretical solutions, as well as practical solutions. Tesseract supports not only real-time cross-chain cryptocurrency trades, but also secure tokenization of assets pegged to cryptocurrencies. For instance, Tesseract-tokenized bitcoins can circulate on the Ethereum blockchain for use in smart contracts. We provide a demo implementation of Tesseract that supports Bitcoin, Ethereum, and similar cryptocurrencies.