The Herschel Multi-tiered Extragalactic Survey (HerMES) has identified large numbers of dusty star-forming galaxies (DSFGs) over a wide range in redshift. A detailed understanding of these DSFGs is hampered by the limited spatial resolution of Herschel. We present 870um 0.45 resolution imaging from the Atacama Large Millimeter/submillimeter Array (ALMA) of 29 HerMES DSFGs with far-infrared (FIR) flux densities in between the brightest of sources found by Herschel and fainter DSFGs found in ground-based sub-millimeter (sub-mm) surveys. We identify 62 sources down to the 5-sigma point-source sensitivity limit in our ALMA sample (sigma~0.2mJy), of which 6 are strongly lensed (showing multiple images) and 36 experience significant amplification (mu>1.1). To characterize the properties of the ALMA sources, we introduce and make use of uvmcmcfit, a publicly available Markov chain Monte Carlo analysis tool for interferometric observations of lensed galaxies. Our lens models tentatively favor intrinsic number counts for DSFGs with a steep fall off above 8mJy at 880um. Nearly 70% of the Herschel sources comprise multiple ALMA counterparts, consistent with previous research indicating that the multiplicity rate is high in bright sub-mm sources. Our ALMA sources are located significantly closer to each other than expected based on results from theoretical models as well as fainter DSFGs identified in the LABOCA ECDFS Submillimeter Survey. The high multiplicity rate and low projected separations argue in favor of interactions and mergers driving the prodigious emission from the brightest DSFGs as well as the sharp downturn above S_880=8mJy.
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