Real-world processes involve objects of different types that interact with each other, like customers, orders, and items. Accurately representing and analyzing these processes requires object-centric process mining techniques that do not force flattening processes to a single perspective. Most of the object-centric modeling, conformance checking, and discovery approaches presented so far capture little to no interaction between objects and their control flow. However, object synchronization is a fundamental necessity to depict such processes accurately. For example, in an order management process, an order should always be confirmed together with all the items it was placed with. In this paper, we propose an object-centric extension to the traditional DECLARE approach for imposing declarative process rules through a set of constraint template instantiations. Through the use of simple primitives, our approach, OC-DECLARE, can express new types of object-centric constraints. Moreover, the pattern-style representation allows also using and interpreting OC-DECLARE constraints individually, e.g., in prediction tasks. We provide a novel algorithm to automatically discover OC-DECLARE constraints from noisy object-centric event logs, including interactions between objects and synchronization behavior. We evaluate the mining approach on four publicly available object-centric datasets, including one real-life one, and identify patterns and constraints that cannot be found using previous object-centric discovery approaches.