Digital Twins are evolving as a key component in modern systems with diverse applications like remote prognostics, optimizing run-time operation, anomaly detection, and more. The essential elements of a digital twin are a virtual representation, a physical asset, and the transfer of data/information between the two. IoT deployments are generally characterized by resource constraints, making synchronization of digital twins with IoT devices more challenging. There is a pressing need to optimize the bandwidth of the data transferred between the system and the twin, while ensuring that the twin is able to capture selected key aspects of the current operational state accurately. In this paper, we present TwinSync, a framework that can be utilized to construct flexible real-time representations of deployed IoT systems and efficiently synchronize relevant system states with the twin, over a communication bottleneck, within a configurable application-specific notion of error (henceforth referred to as approximate synchronization). Our approach is optimized to achieve data transfers utilizing less bandwidth without compromising the ability of the twin to replicate real-time system states within the specified approximate synchronization semantics. We evaluate the efficacy of TwinSync's synchronization by conducting both a synthetic analysis and a case study based on a real-life application prototype. Our evaluation indicates that using TwinSync can provide the same or greater accuracy (in many cases) while sending significantly fewer bytes than a bandwidth-insensitive synchronization approach. The result is attributed to a more judicial selection of data to transmit over bottlenecks, compared to bandwidth-insensitive approaches.