Several thousand eclipsing binaries (EBs) have been found in photometric surveys of the Magellanic Clouds. Periods, amplitudes, and other characteristics of these variable stars have been determined mostly from automatic pipelines, and accurate parameters of the eclipsing pairs and their components are only known for a small fraction of the EBs, mostly for detached ones. We focus on a group of stars in the near contact stage, launching a new project to the improve the accuracy of known parameters and to study any (dis)similarities among systems in our Galaxy and in the Magellanic Clouds.