Final Outcome - Second shipment arrived before the second re-ship was needed.
The package had no indication of its contents, the sender or the transaction involved. The label was a computer-printed label that was 100% correct.
I have no idea why both packages - the first, which did not arrive, and the second which did - have entries in the tracking history which includes “Undeliverable as Addressed - Return to Sender.” As far as I know, this has not happened before in the ~10 years I’ve lived at this address.
ILGM was perfectly nice about it, and the only thing I’d improve if I could was to decrease the amount of time it takes for ILGM to actually ship something (it takes a few days between “order placed” and “order shipped”).