Optical burst switching (OBS) is able to operate under both asynchronous and synchronous burst departures. Although the asynchronous approach is generally assumed in OBS owing to its simplicity and low technical requirements, it has been shown that performance improvements, in terms of the overall burst loss probability, can be achieved under synchronous operation. Moreover, when effective contention resolution mechanisms are applied, such improvements are significant. Since perfect synchronization is not a viable solution, we propose a novel operation mode with relaxed technical requirements, the quasi-synchronous (QS) operation. In this paper, the performance of the QS operation mode is evaluated by an exact analytic expression, assuming a Poisson burst arrival process and a single-wavelength scenario. This mathematical analysis is completed with numerical results that validate its accuracy. Furthermore, we tackle the issue of not having well-aligned clock information between network nodes by proposing a novel re-synchronization mechanism for OBS networks. Finally, a performance study of the QS operation under different deflection routing techniques is presented. The results show that our novel QS approach achieves significant performance benefits with respect to the asynchronous operation while avoiding the architectural complexity of the synchronous scheme.