Making Native WebDAV Actually Work on nginx with Finder and Explorer

So my long march away from Apache has been coming to an end, and I am finally migrating some of the more esoteric parts of my Apache setup to nginx. I have a side domain that I use to share files with some friends and, for ease of use, I have configured it with WebDAV so that they can simply mount it using Finder or Explorer, just like a shared drive. The problem? nginx’s WebDAV support … sucks. First, the ngx_http_dav_module module is not included in most distributions from the package managers. Even the ones that are, it’s usually pretty out of date. And, perhaps worst of all, it is a partial implementation of WebDAV. It doesn’t support some of the things (PROPFIND, OPTIONS, LOCK, and UNLOCK) that are needed to work with modern clients. So what can we do?