An update: I got frustrated with getting this working in AIX, and so instead moved to a Linux box (which is a less-optimal environment for me to work in for reasons I won't get into here) and reinstalled a newer version of Perl there. However, Net::OpenSSH ended up failing with almost an identical error! "muxserver_listen bind(): Operation not permitted".

Since this is a more common error, a Google search turned up more information. It turns out that, as salva guessed earlier, the unix socket could not be created under ~/.libnet-openssh-perl. However, the problem wasn't a permission one but, since my home directory is in AFS, AFS doesn't permit the creation of sockets.

So I'm now trying the following:

my $ssh = Net::OpenSSH->new('username@hostname', strict_mode => 0, ctl_dir => "/tmp/.libnet-openssh-perl");

...and this works! Of course, I'm running without strict mode, which is supposedly insecure (which I don't understand since if .libnet-openssh-perl prohibits others from writing to the directory, what good forcing all ancestors to also prohibit writing to their directory gains, but I'm not a security expert). But it's either that or not run Net::OpenSSH at all since the AFS directories exist (at least in part) so I don't need to be given write access to anywhere local on the box (besides /tmp, which gives write access to everyone).