It turns out that, out of the four possible permutations of including or excluding trailing or leading forward slashes on the BaseAddress and the relative URI passed to the GetAsync method — or whichever other method of HttpClient — only one permutation works. You must place a slash at the end of the BaseAddress, and you must not place a slash at the beginning of your relative URI, as in the following example.

using (var handler = new HttpClientHandler())
using (var client = new HttpClient(handler))
{
    client.BaseAddress = new Uri("http://something.com/api/");
    var response = await client.GetAsync("resource/7");
}

Even though I answered my own question, I figured I’d contribute the solution here since, again, this unfriendly behavior is undocumented. My colleague and I spent most of the day trying to fix a problem that was ultimately caused by this oddity of HttpClient.