We present images and light curves of the bipolar planetary nebula RPJ 053059-683542 that was discovered in the Reid-Parker AAO UKST Hα survey of the Large Magellanic Cloud. The emission from this object appears entirely nebular, with the central star apparently obscured by a central band of absorption that bisects the nebula. The light curves, which were derived from images from the Super MACHO project at CTIO, showed significant, spatially resolved variability over the period 2002 January through 2005 December. Remarkably, the emission from the two bright lobes of the nebula vary either independently or similarly but with a phase lag of at least 1 yr. The optical spectra show a low level of nebular excitation, and only modest N enrichment. Infrared photometry from the 2MASS and SAGE surveys indicates the presence of a significant quantity of dust. The available data imply that the central star has a close binary companion and that the system has undergone some kind of outburst event that caused the nebular emission to first brighten and then fade. Further monitoring, high-resolution imaging, and detailed IR polarimetry and spectroscopy would uncover the nature of this nebula and the unseen ionizing source.