Aims. AX J0043−737 is a source in the ASCA catalogue whose nature is uncertain. It is most commonly classified as a Crab-like pulsar in the Small Magellanic Cloud (SMC) following apparent detection of pulsations at ~87 ms from a single ASCA observation. A follow-up ASCA observation was not able to confirm this, and the X-ray detection of the source has not been reported since. Methods. We studied the nature of the source with a dedicated XMM-Newton observation. We ascertained the source position, searched for the most probable counterpart, and studied the X-ray spectrum. We also analysed other archival observations with the source in the field of view to study its long-term variability. Results. With the good position localisation capability of XMM-Newton, we identify the counterpart of the source as MQS J004241.66–734041.3, an active galactic nucleus (AGN) behind the SMC at a redshift of 0.95. The X-ray spectrum can be fitted with an absorbed power law with a photon-index of Γ = 1.7, which is consistent with that expected from AGNs. By comparing the current XMM-Newton observation with an archival XMM-Newton and two other ASCA observations of the source, we find signatures of long-term variability, another common phenomenon in AGNs. All of the above are consistent with AX J0043−737 being an AGN behind the SMC.