Following up mass surveillance and privacy issues, modern secure communication protocols now seek more security such as forward secrecy and post-compromise security. They cannot rely on an assumption such as synchronization, predictable sender/receiver roles, or online availability. Ratcheting was introduced to address forward secrecy and post-compromise security in real-world messaging protocols. At CSF 2016 and CRYPTO 2017, ratcheting was studied either without zero round-trip time (0-RTT) or without bidirectional communication. At CRYPTO 2018, ratcheting with bidirectional communication was done using heavy key-update primitives. At EUROCRYPT 2019, another protocol was proposed. All those protocols use random oracles. Furthermore, exchanging n messages has complexity O(n2). In this work, we define the bidirectional asynchronous ratcheted key agreement (BARK) with formal security notions. We provide a simple security model and design a secure BARK scheme using no key-update primitives, no random oracle, and with O(n) complexity. It is based on a cryptosystem, a signature scheme, one-time symmetric encryption, and a collision-resistant hash function family. We further show that BARK (even unidirectional) implies public-key cryptography, meaning that it cannot solely rely on symmetric cryptography.