Can you please describe (or show relevant code) of your setup in a bit more detail just so we can understand your situation better?

I think your setup is something like the following:

local machine                remote machine
script         Net::OpenSSH  fastCGI script
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... and you want to run the fastCGI script and get its output?

Maybe you can also show us how you're starting the fastCGI script exactly. Maybe FastCGI is happier with having actual filehandles (with inodes), or it wants a /dev/tty or a domain socket to interact with, instead what Net::OpenSSH provides directly.

A nasty alternative could be to set up just enough of Apache (or another FastCGI environment using Net::FastCGI or whatever), and then access the script via http...

I mean I have an apache application, using CGI::Fast (which uses FCGI). This application is working happily.

Now I'm trying to add a request handler that upon receiving a request via apache, it will attempt to use Net::OpenSSH to connect to a remote host

The issue is FCGI / CGI::Fast have captured and redirected the STDIN/STDOUT/STDERR filehandles and Net::OpenSSH dies when I try to create a new Net::OpenSSH object

Abbreviated code below

use CGI::Fast;
while (my $r = CGI::Fast->new) {
        open my $stderr_fh, '>', 'some_file';
        open my $stdin_fh, '<', '/dev/null';
        open my $stdout_fh, '>', 'some_other_file';

        my %options = (
                batch_mode => 1,
                timeout => 10,
                kill_ssh_on_timeout => 1,
                default_stdout_discard => 1,
                default_stderr_discard => 1,
                master_stderr_fh => $stderr_fh,
                master_stdout_fh => $stdout_fh,
        );

        my $ssh = Net::OpenSSH->new('test@192.168.0.123', %options);
}
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The errors invariably occur inside the Net::OpenSSH->new() call

Try with default_stdin_fh => $stdin_fh.

Also, enable debugging in order to see what is happening: $Net::OpenSSH::debug=127;.