Nginx gotcha

Yesterday morning I noticed some odd behavior: Nginx was serving pages with the wrong mime type. All of a sudden, links to cached pages were interpreted as octet-stream, when instead they were just straight-up html.

I suspected immediately that it had something to do with Rails page caching with Passenger plus Nginx, but what? Rails caches pages with the correct file extension, so shouldn't Nginx pick up on that and serve the page correctly? Turns out it's more complicated than that.