Quoting: AnxiousInfusionYou gave me the idea to chase the 32 bit mesa packages down the backported rabbit hole.

That was my point. According to debian steam package, only libgl1-mesa-dri and libgl1-mesa-glx are available on backports repos. But yeah, looking for every mesa packages makes sense. I have a Nvidia GPU, so I don't know much about mesa drivers.

Quoting: slaapliedjeSo, being the Debian Connoisseur that I am... Synaptic doesn't really support multi-arch worth a damn. Sad, I know, but it doesn't.

I didn't have any problem with synaptic and multi-arch. Only with backports.
I think the "Force package version" uses the following syntax:
apt-get install libgl1-mesa-dri:i386/jessie-backports
which means: get this one package from backports and then try to install it with dependencies from stable repos. Thus leads to a possible broken system.

Quoting: slaapliedjehttps://packages.debian.org/search?keywords=steam&searchon=names&suite=all&section=all shows that there is no steam in jessie-backports, just use regular ol' Jessie

As said in manpage, the point of -t jessie-backports is to set a high priority to backports repos, and then fallback to stable ones. So backports repos are first checked, then if nothing available, check stable repos. A sort of one-shot pinning for this one command. Since steam needs specific driver dependencies and AnxiousInfusion has backported drivers, I thought it's much safe to enable backports repos and prioritize them over stable ones.

Instead of creating a new topic, I'm back with related issues:

I was very excited to find updates for my Mesa driver enter Debian backports the other day. So I immediately upgraded from 11.1 to 12 and probably made a mistake by going through Synaptic package manager instead of apt. At first, all but two mesa packages upgraded due to held/broken. After a quick apt autoremove, I tried again and the remaining "libgl-mesa-dri" packages upgraded. One reboot later and Cinnamon is now running in software rendering mode. I think it dun broke.

Does Mesa 12 require a recent version of X11? Are there other packages needed for Mesa 12's new additions of Vulkan.. etc.?

I got my system broken as well. I got a first batch of mesa 12 libs 2 days ago, did the update and everything was ok. I got a new batch of mesa 12 libs yesterday, did the update and it broke everything that is related to OpenGL: steam, games, wine, mpv and KDE too.

I finally uninstalled and re-installed the Nvidia prop drivers, and it fixed everything.

Yours is a different story. Try to look at this post:
News for package mesa: Accepted mesa 12.0.3-1~bpo8+1 (source amd64) into jessie-backports
and check out if every new libs are installed in your system, amd64 + i386.

Since I don't know much about pure mesa, I won't be very helpful.

The dependencies are all met (both amd64 and i386). When I search for any information on the topic, the only results are... this thread. This means either that I'm the only person trying to do this or that everyone else who has done this already know the secrets to get it working and therefor has no need to talk about it.

EDIT 11/27: Turns out, Mesa 12.0.3 backport was packaged incorrectly. Installation of libgbm1 and an upgrade to 12.0.4 has resolved the issue.