Opauth is a multi-provider authentication framework for PHP, inspired by OmniAuth for Ruby.
Opauth enables PHP applications to do user authentication with ease.
Try out Opauth for yourself at http://opauth.org.
Fuel-Opauth is just a simple package for Opauth. https://github.com/andreoav/fuel-opauth
I haven't used it, but looking at the code it expects the stategy to be used as the first segment in the URI.
If you haven't got your rewrites properly done, your URI has /public/ as first segment, which is not going to work. I expect this is the case, since the error message mentions the given strategy 'public'.