Multi-agent multi-target tracking has a wide range of applications, including wildlife patrolling, security surveillance or environment monitoring. Such algorithms often assume that agents are pre-assigned to monitor disjoint partitions of the environment, reducing the burden of exploration. This limits applicability when there are fewer agents than targets, since agents are unable to continuously follow the targets in their fields of view. Multi-agent tracking algorithms additionally assume a central controller and synchronous inter-agent communication. Instead, we focus on the setting of decentralized multi-agent, multi-target, simultaneous active search-and-tracking with asynchronous inter-agent communication. Our proposed algorithm MASTER uses a sequential monte carlo implementation of the probability hypothesis density filter for posterior inference combined with Thompson sampling for decentralized multi-agent decision making. We compare different action selection policies, focusing on scenarios where targets outnumber agents. In simulation, MASTER outperforms baselines in terms of the Optimal Sub-Pattern Assignment (OSPA) metric for different numbers of targets and varying teamsizes.