Note that there are various cleanups that can be made if we keep
this change, but I do not want to start making changes that
depend on this one until the 1.4 cycle closes.

Fixes #8405.

LGTM=r
R=golang-codereviews, adg, r, bradfitz
CC=golang-codereviews, iant
https://golang.org/cl/130240043