In a search for the counterpart to the Fermi-LAT source 3FGL J0838.8-2829, we performed a multiwavelength campaign: in the X-ray band with Swift and XMM-Newton; in the infrared and optical with OAGH, ESO-NTT and IAC80; and in the radio with ATCA observations. We also used archival hard X-ray data obtained by INTEGRAL. We report on three X-ray sources consistent with the position of the Fermi-LAT source.We confirm the identification of the brightest object, RX J0838-2827, as a magnetic cataclysmic variable that we recognize as an asynchronous system (not associated with the Fermi-LAT source). RX J0838-2827 is extremely variable in the X-ray and optical bands, and timing analysis reveals the presence of several periodicities modulating its X-ray and optical emission. The most evident modulations are interpreted as being caused by the binary system orbital period of ~1.64 h and the white dwarf spin period of ~1.47 h. A strong flux modulation at ~15 h is observed at all energy bands, consistent with the beat frequency between spin and orbital periods. Optical spectra show prominent Hß, He I and He II emission lines that are Doppler-modulated at the orbital period and at the beat period. Therefore, RX J0838-2827 accretes through a disc-less configuration and could be either a strongly asynchronous polar or a rare example of a pre-polar system on its way to reaching synchronism. Regarding the other two X-ray sources, XMM J083850.4-282759 showed a variable X-ray emission, with a powerful flare lasting for ~600 s, similar to what is observed in transitional millisecond pulsars during the subluminous disc state: this observation possibly means that this source can be associated with the Fermi-LAT source.