Context. Over half of the cometary nuclei and a part of the asteroids that have been photographed so far by space missions or imaged by Doppler radar techniques appear to be bilobate or contact binary systems. The latest research on these objects shows that rotational fission and fragment reconnection can lead to reconfiguration, creating the next generation of bilobate bodies. In this context, Main-belt comet 288P, the only known double object of this class with components of comparable masses, appears to have successfully avoided reconfiguration or disassociation into a dynamically unbound pair and has become a wide asynchronous binary. Aims. Our goal was to determine the physical parameters, such as sizes, shapes, and rotation periods, of both components of 288P to understand how this double asteroid formed and how it has evolved to obtain today’s very wide orbit. We also tried to confirm or deny the existence of a third component in a tight pair with the larger, slowly rotating fragment, as previously suggested. Methods. We obtained a composite light curve of 288P by observing this object with the Gemini South and Keck II telescopes working in tandem. Through model analysis we separated this light curve into components, one for each fragment. We found their sidereal rotation periods and the most probable shapes and sizes. We analysed the angular momentum and energy balances and compared actual values with that expected at the moment of rotational splitting to check how much surplus has been introduced into the system. Results. We determined the rotation periods of the components to be 15.86 hours for the larger object A and 3.37 hour for the smaller fragment B. Assuming a geometric albedo of 0.07 in the R photometric band, surface and reflectance properties adequate for C-type asteroids and comets, and considering A and B as prolate spheroids, we found that their semi-axes a, b (where b < a) are equal to 1.12, 0.69 and 0.67, 0.57 km for the larger and smaller components, respectively. The existence of a third body in 288P cannot be definitely excluded but should be considered as unlikely. Conclusions. A plausible mechanism responsible for the origin of the binary asteroid 288P is rotational fission of a bilobate progenitor spun up by the Yarkovsky-O’Keefe-Radzievskii-Paddack mechanism or, more likely, by sublimation-driven torque produced by an active region or regions. It is almost certain that the sublimative activity of the smaller fragment B is behind its relatively fast, completely asynchronous rotation and the wide mutual orbit of the components.