Authentication of Apache+SVN server behind nginx reverse proxy

I have the same issue, and I got it solved by using the apache mod_authn_anon module.

Apache configuration:

LoadModule authn_anon_module /usr/lib/apache2/mod_authn_anon.so
LoadModule dav_svn_module /usr/lib/apache2/mod_dav_svn.so
LoadModule authz_svn_module /usr/lib/apache2/mod_authz_svn.so

<Location /some/path/svn>
        DAV svn
        SVNPath /svn
        SVNReposName /some/path/svn

        # Authentication is performed by the reverse proxy, the username is
        # forwarded/set in the Remote-User header by the proxy.
        # Use anon authentication module, allowing anybody to access this path,
        # but registering the user in svn.
        AuthType Basic
        AuthName SVN
        AuthBasicProvider anon
        Anonymous "*"
        Require valid-user
</Location>

Nginx configuration:

 location /some/path/svn {
    proxy_set_header Host $host;
    proxy_pass http://svn/some/path/svn;
    proxy_set_header Remote-User $remote_user;
}