We report on multiwavelength observations of the blazar PKS 0537-441 (z = 0.896) obtained from microwaves through gamma rays by SMA, REM, ATOM, Swift and Fermi during 2008 August-2010 April. Strong variability has been observed in gamma rays, with two major flaring episodes (2009 July and 2010 March) and a harder-when-brighter behaviour, quite common for FSRQs and low-synchrotron-peaked BL Lacs, in 2010 March. In the same way the SED of the source cannot be modelled by a simple synchrotron self-Compton model, as opposed to many BL Lacs, but the addition of an external Compton component of seed photons from a dust torus is needed. The 230 GHz light curve showed an increase simultaneous with the gamma-ray one, indicating co-spatiality of the mm and gamma-ray emission region likely at large distance from the central engine. The low, average, and high activity SED of the source could be fit changing only the electron distribution parameters, but two breaks in the electron distribution are necessary. The ensuing extra spectral break, located at NIR-optical frequencies, together with that in gamma rays seem to indicate a common origin, most likely due to an intrinsic feature in the underlying electron distribution. An overall correlation between the gamma-ray band with the R-band and K-band has been observed with no significant time lag. On the other hand, when inspecting the light curves on short time scales some differences are evident. In particular, flaring activity has been detected in NIR and optical bands with no evident gamma-ray counterparts in 2009 September and November. Moderate variability has been observed in X-rays with no correlation between flux and photon index. An increase of the detected X-ray flux with no counter part at the other wavelengths has been observed in 2008 October, suggesting once more a complex correlation between the emission at different energy bands. ; Comment: 12 pages, 10 figures, 3 tables, MNRAS in press - Minor editing changes