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 Intel SGX as a trusted execution environment. Furthermore, by running a consensus protocol among SGX-enabled servers, Tesseract mitigates denial-of-service attacks. 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 reference implementation of Tesseract that supports Bitcoin, Ethereum, and similar cryptocurrencies.