  Very faint X-ray transients (VFXTs) are X-ray transients with peak X-ray
luminosities ($L_X$) of $L_X\lesssim10^{36}$ erg/s, which are not
well-understood. We carried out a survey of 16 square degrees of the Galactic
Bulge with the Swift Observatory, using short (60 s) exposures, and returning
every 2 weeks for 19 epochs in 2017-18 (with a gap from November 2017 to
February 2018, when the Bulge was in sun-constraint). Our main goal was to
detect and study VFXT behaviour in the Galactic Bulge across various classes of
X-ray sources. In this work, we explain the observing strategy of the survey,
compare our results with the expected number of source detections per class,
and discuss the constraints from our survey on the Galactic VFXT population. We
detected 91 X-ray sources, 25 of which have clearly varied by a factor of at
least 10. 45 of these X-ray sources have known counterparts: 17
chromospherically active stars, 12 X-ray binaries, 5 cataclysmic variables (and
4 candidates), 3 symbiotic systems, 2 radio pulsars, 1 AGN, and a young star
cluster. The other 46 are of previously undetermined nature. We utilize X-ray
hardness ratios, searches for optical/infrared counterparts in published
catalogs, and flux ratios from quiescence to outburst to constrain the nature
of the unknown sources. Of these 46, 7 are newly discovered hard transients,
which are likely VFXT X-ray binaries. Furthermore, we find strong new evidence
for a symbiotic nature of 4 sources in our full sample, and new evidence for
accretion power in 6 X-ray sources with optical counterparts. Our findings
indicate that a large subset of VXFTs is likely made up of symbiotic systems.
