How to autocomplete file paths in Vim, just like in zsh?

In Zsh, I can use filename completion with slashes to target a file deep in my source tree. For instance if I type:

 vim s/w/t/u/f >TAB<

zsh replaces the pattern with:

 vim src/wp-contents/themes/us/functions.php

What I'd like is to be able to target files the same way at the Vim command line, so that typing

 :vi s/w/t/u/f >TAB<

will autocomplete to:

 :vi src/wp-contents/themes/us/functions.php

I'm trying to parse the Vim docs for wildmode, but I don't see what settings would give me this. It's doing autocompletion for individual filenames, but not file paths. Does Vim support this natively? Or how can I customize the autocomplete algorithm for files?

Thanks for any advice! -mykle-