After a lot of trial and error fiddling then the following PerlApp line gets it to work:

perlapp -v -r -s=alwayson.pl -f -a=AutoLoader;Config;DynaLoader;Errno;
+IO;LWP;SelectSaver;Socket;Symbol;URI;XSLoader;Exporter::Heavy;IO::Han
+dle;IO::Select;IO::Socket;IO::Socket::INET;IO::Socket::UNIX;LWP::Prot
+ocol::HTTP;URI::_FOREIGN;URI::_GENERIC;URI::_QUERY;URI::_SERVER;URI::
+URL;URI::HTTP
[download]

But how to arrive at this list of modules *without* the nasty trial and error part? Anybody know?