I can't say about anything specific for the tournament, but there is not much of a permanent gopher fan base in DC. I believe Front Page (near DuPont Circle) is supposed to be a "Gopher Bar", but I never scoped it out. I only lived a few blocks away last year, but I couldn't stomach watching that team in public. I just moved back to the cities but can offer some suggestions based on what neighborhood you are staying in and what scene you are looking for.
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