what benefit do you expect to derive from removing 'use strict;' from your production code?

However, since the release our customers have uncovered a few more errors (always because of symbolic references). I suspect more are lurking, and because we don't have any automated testing it's not likely that we'll find every combination of settings/inputs that cause these errors. That means our customers will find the problems, which is bad.

I can understand where it would be bad for a customer to find a bug, but IMO it's also bad for that customer NOT to receive an error, and to continue on as if nothing happened when in reality, it is probable that data has been lost, or there has been some other problem with their transaction.
If you give a man a fish he will eat for a day.
If you teach a man to fish he will buy an ugly hat.
If you talk about fish to a starving man, you're a consultant.

What is actually funny about this, is that there are IT managers, in FINANCIAL INSTITUTIONS in Australia no less, that demand error checking be removed from code so stuff "just works" (or appears to).

I was once berated for reporting an error after receiving error codes from libraries written by my very manager. After that experience I promptly removed the error checking (as he wanted) and resigned.

Whilst I agree it is a greater sin to err in silence, there's significant IT systems supporting financial infrastructures that hang together only on ego. Just shocking.