Very faint X-ray transients (VFXTs) are X-ray transients with peak X-ray luminosities ($L_X$) of $L_Xlesssim10^{36}$ 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.
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