ABSTRACT BG Ind is a well-studied, bright, nearby binary consisting of a pair of F stars in a 1.46-d orbit. We have discovered in the TESS light curve for TIC 229804573 (aka BG Ind), a second eclipsing binary in the system with a 0.53-d period. Our subsequent analyses of the recent TESS and archival ground-based photometric and radial velocity (RV) data reveal that the two binaries are gravitationally bound in a 721-d period, moderately eccentric orbit. We present the results of a joint spectro-photodynamical analysis of the eclipse timing variation curves of both binaries based on TESS and ground-based archival data, the TESS light curve, archival RV data, and the spectral energy distribution, coupled with the use of PARSEC stellar isochrones. We confirm prior studies of BG Ind that found that the brighter binary A consists of slightly evolved F-type stars with refined masses of 1.32 and 1.43 M⊙, and radii of 1.59 and 2.34R⊙. The previously unknown binary B has two less massive stars of 0.69 and 0.64 M⊙ and radii of 0.64 and 0.61R⊙. Based on a number of different arguments that we discuss, we conclude that the three orbital planes are likely aligned to within 17○.