This was originally done in https://golang.org/cl/31359 but partially
undone (apparently unintentionally) in https://golang.org/cl/34310

Fix it, and update tests to ensure the error is unrecoverable.

Fixes #23039

Change-Id: I923ebd613a05e67d8acce77f4a68c64c8574faa6
Reviewed-on: https://go-review.googlesource.com/82656
Run-TryBot: Brad Fitzpatrick <bradfitz@golang.org>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-by: Dmitry Vyukov <dvyukov@google.com>