Multi-agent multi-target tracking has a wide range of applications, including wildlife patrolling, security surveillance or environment monitoring. Such algorithms often make restrictive assumptions: the number of targets and/or their initial locations may be assumed known, or agents may be pre-assigned to monitor disjoint partitions of the environment, reducing the burden of exploration. This also 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 inter-agent synchronization of observations, or the presence of a central controller to coordinate joint actions. 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 DecSTER 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, we demonstrate that DecSTER is robust to unreliable inter-agent communication and outperforms information-greedy baselines in terms of the Optimal Sub-Pattern Assignment (OSPA) metric for different numbers of targets and varying teamsizes.