broquaint, thanks.
I am aware of Parallel::ForkManager, but I figured it wasn't worth the grief of installing an extra module for a relatively small amount of code.

Of course, that was before I realised my code wasn't playing nice...

Update: I've tried broquaint's suggestion of Parallel::ForkManager, and that suffers from the same problem on Solaris (5.6.1 and 5.8.4), but is fine on Linux.

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.