We consider the problem of cooperative exploration where multiple robots need to cooperatively explore an unknown region as fast as possible. Multi-agent reinforcement learning (MARL) has recently become a trending paradigm for solving this challenge. However, existing MARL-based methods adopt action-making steps as the metric for exploration efficiency by assuming all the agents are acting in a fully synchronous manner: i.e., every single agent produces an action simultaneously and every single action is executed instantaneously at each time step. Despite its mathematical simplicity, such a synchronous MARL formulation can be problematic for real-world robotic applications. It can be typical that different robots may take slightly different wall-clock times to accomplish an atomic action or even periodically get lost due to hardware issues. Simply waiting for every robot being ready for the next action can be particularly time-inefficient. Therefore, we propose an asynchronous MARL solution, Asynchronous Coordination Explorer (ACE), to tackle this real-world challenge. We first extend a classical MARL algorithm, multi-agent PPO (MAPPO), to the asynchronous setting and additionally apply action-delay randomization to enforce the learned policy to generalize better to varying action delays in the real world. Moreover, each navigation agent is represented as a team-size-invariant CNN-based policy, which greatly benefits real-robot deployment by handling possible robot lost and allows bandwidth-efficient intra-agent communication through low-dimensional CNN features. We first validate our approach in a grid-based scenario. Both simulation and real-robot results show that ACE reduces over 10% actual exploration time compared with classical approaches. We also apply our framework to a high-fidelity visual-based environment, Habitat, achieving 28% improvement in exploration efficiency.