  BG Ind is a well studied, bright, nearby binary consisting of a pair of F
stars in a 1.46-day orbit. We have discovered in the TESS lightcurve for TIC
229804573 (aka BG Ind) a second eclipsing binary in the system with a 0.53-day.
Our subsequent analyses of the recent TESS and archival ground-based
photometric and radial velocity data, reveal that the two binaries are
gravitationally bound in a 721-day 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 lightcurve, archival radial velocity data and the
spectral energy distribution, coupled with the use of PARSEC stellar
isochrones. We confirm prior studies of BG Ind which found that the brighter
binary A consists of slightly evolved F-type stars with refined masses of 1.32
and 1.43 $M_\odot$, and radii of 1.59 and 2.34 $R_\odot$. The previously
unknown binary B has two less massive stars of 0.69 and 0.64 $M_\odot$ and
radii of 0.64 and 0.61 $R_\odot$. Based on a number of different arguments
which we discuss, we conclude that the three orbital planes are likely aligned
to within 17$^\circ$.
