14.04 -> 16.04 upgrade, broken libraries, broken libc

Question

I had the same symptoms as in this question, and took the suggested remedy of installing the libstdc++ from here. After that I got a ton of complaints about chromium and libreoffice that I could only resolve with dpkg -r (apt-get remove didn't work), and trying to use apt-get I still get these errors:

perec ~/Downloads 10:10:48 $ sudo apt-get install apt
Reading package lists... Done
Building dependency tree
Reading state information... Done
apt is already the newest version (1.2.19).
You might want to run 'apt-get -f install' to correct these:
The following packages have unmet dependencies:
 libstdc++-6-dev : Depends: libstdc++6 (>= 6.1.1-3ubuntu11~14.04.1) but 5.4.0-6ubuntu1~16.04.4 is to be installed
 libstdc++6 : Depends: gcc-5-base (= 5.4.0-6ubuntu1~16.04.4) but it is not going to be installed
 libstdc++6:i386 : Depends: gcc-5-base:i386 (= 5.4.0-6ubuntu1~16.04.4) but it is not going to be installed
                   Depends: libc6:i386 (>= 2.18) but it is not going to be installed
                   Depends: libgcc1:i386 (>= 1:4.2) but it is not going to be installed
E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution).

Ok, what if I try apt-get -f install?

perec ~/Downloads 10:10:51 $ sudo apt-get -f install
Reading package lists... Done
Building dependency tree
Reading state information... Done
Correcting dependencies... failed.
The following packages have unmet dependencies:
 libstdc++-6-dev : Depends: libstdc++6 (>= 6.1.1-3ubuntu11~14.04.1) but 5.4.0-6ubuntu1~16.04.4 is installed
 libstdc++6 : Depends: gcc-5-base (= 5.4.0-6ubuntu1~16.04.4) but it is not installed
 libstdc++6:i386 : Depends: gcc-5-base:i386 (= 5.4.0-6ubuntu1~16.04.4) but it is not installed
                   Depends: libc6:i386 (>= 2.18) but it is not installed
                   Depends: libgcc1:i386 (>= 1:4.2) but it is not installed
E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.
E: Unable to correct dependencies

Hooray!

Incidentally, this ticket suggests installing both the amd64 and i386 libstdc++ .deb files, for some reason:

perec ~/Downloads 10:13:24 $ sudo dpkg -i libstdc++6_5.4.0-6ubuntu1~16.04.4_*
(Reading database ... 187526 files and directories currently installed.)
Preparing to unpack libstdc++6_5.4.0-6ubuntu1~16.04.4_amd64.deb ...
Unpacking libstdc++6:amd64 (5.4.0-6ubuntu1~16.04.4) over (5.4.0-6ubuntu1~16.04.4) ...
Preparing to unpack libstdc++6_5.4.0-6ubuntu1~16.04.4_i386.deb ...
Unpacking libstdc++6:i386 (5.4.0-6ubuntu1~16.04.4) over (5.4.0-6ubuntu1~16.04.4) ...
dpkg: dependency problems prevent configuration of libstdc++6:amd64:
 libstdc++6:amd64 depends on gcc-5-base (= 5.4.0-6ubuntu1~16.04.4); however:
  Package gcc-5-base is not installed.

dpkg: error processing package libstdc++6:amd64 (--install):
 dependency problems - leaving unconfigured
dpkg: dependency problems prevent configuration of libstdc++6:i386:
 libstdc++6:i386 depends on gcc-5-base (= 5.4.0-6ubuntu1~16.04.4).
 libstdc++6:i386 depends on libc6 (>= 2.18).
 libstdc++6:i386 depends on libgcc1 (>= 1:4.2).

dpkg: error processing package libstdc++6:i386 (--install):
 dependency problems - leaving unconfigured
Processing triggers for libc-bin (2.23-0ubuntu5) ...
Errors were encountered while processing:
 libstdc++6:amd64
 libstdc++6:i386

Of course, I can't install gcc-5-base, etc., using apt-get, because, well, it considers my system broken. And I have no idea if I even should install those things.

(Wonderfully, I just quit pidgin and now can't restart it, because the binary was apparently deleted in the upgraded. And I can't reinstall it, either, because apt-get is broken. Swell upgrade process here.)