Ports are striving for innovative technological solutions to cope with the increasing growth in demand of goods transport, while at the same time improving their environmental footprint. An emerging technology that has the potential to substantially increase the effectiveness of the multifaceted and interconnected port processes is that of digital twins. Recognizing the potential of twinning assets and processes, innovation-leading ports have already started working on it. However, since there is no clear consensus on what a digital twin of a complex system comprises and how it should be designed, deployed digital twin solutions for ports, as in multiple other domains, often differ significantly. We address this issue by initially identifying three core aspect underpinning digital twins of complex systems, grounding our work on the domain of ports, and outlining five successive maturity levels based on these aspects' instantiation. These identified aspects and the derived maturity levels are then used to examine real-world cases by critically evaluating existing digital twinning solutions in the port of Singapore, the Mawan port of Shanghai, and that of Rotterdam. Our work on maturity levels and core twinning aspects can provide a guideline for designing and benchmarking digital twinning solutions, applicable in any domain where a digital twin can be used. For ports, the capacity for innovation via twinning is highly contextual, with the key affecting factors being the availability of financial and technical resources.