Of course, that was before I realised my code wasn't playing nice...
Update 2: I tried another tactic - don't read the configuration file and fork children in a single loop.
Instead, I open and read the configuration file and store the information in an array, then iterate over the array using foreach, kicking off children and reaping as before.
This approach does work correctly, but still doesn't reveal why the original code fails.
If the information in this post is inaccurate, or just plain wrong, don't just downvote - please post explaining what's wrong.
That way everyone learns.