What is Multiarch?
In order to ease cross-compiling and distribution, the Debian-based distros have undertaken a project known as "multi-arch." The idea is to update the packaged software itself, and occasionally change how it's built, so that every package can satisfy multiple architectures. Either one version of the package will work for all architectures, or the different architectures' versions can be installed alongside each other.
To be fully compliant, a package not only needs to play well with different architectures, but explicitly state how it operates in a multi-arch environment. This is implemented through values in the control file of each source package, which are then enforced by the distro's package management.
While there are general guidelines, each package has its own dynamic so making it multi-arch compatible is largely up to that package's maintainer. As a result, multi-arch is still a work in progress, and the distros prioritize which packages they focus on.
According to documents from ca. 2013, multi-arch behavior for header-files (common in -dev packages) was undefined. Since then, the Debian-family distros have settled on using subdirectories for multi-arch triplets within /usr/include/. Note that this extension is not yet part of the Filesystem Hierarchy Standard.
Status of Wine Build Dependencies
Perhaps the main (but not only) way this matters to Wine is in regards to building and porting. Once all of Wine's build dependencies are multi-arch compatible, cross-compiling and packaging Wine for different architectures becomes much simpler.
As of August 2017, building Wine 2.14 on Debian Stretch shows the following build dependencies still conflicting across architectures (version info may be different for Ubuntu, Mint, etc.)
|Debian Package||Stable Version||Unstable Version||Apt-get Result||Debian Bug Report||Patch Proposed?||Fixed in Stretch?||Fixed in Buster?||Notes|
|libxml-simple-perl||2.22-1||2.24-1||Package couldn't be found||✔||✔||Outdated, only used once in dlls/opengl32/make_opengl; consider move to libxml-libxml-simple-perl... or rewrite make_opengl for new tools|
|oss4-dev||4.2-build2010-2||4.2-build2010-5||Package couldn't be found||Necessary for OSS sound-driver support in Wine. Replaced by ALSA.|
|khronos-api||0~svn29735-1.1||0~svn33340-0.1||Package couldn't be found||✔||✔||Supersedes parts of dlls/opengl32/make_opengl|
|libpcap-dev||1.8.1-3||1.8.1-3||Package couldn't be found||#760370||✔||Transitional package; pulls in libpcap0.8-dev (not yet multi-arch)|
|gir1.2-gstreamer-1.0||1.4.4-2||1.6.3-1||Satisfies dependencies, but won't be installed||Needed by libgstreamer1.0-dev; see #755874 about multiarch in gobject-introspection|
|freeglut3-dev||2.8.1-2||2.8.1-2||Other arch will be removed||N/A||N/A||✔||N/A||Depends on libglu1-mesa-dev|
|libatk1.0-dev||2.14.0-1||2.18.0-1||Other arch will be removed||#689124||✔||Also depends on libglib2.0-dev|
|libavahi-client-dev||0.6.31-5||0.6.32~rc+dfsg-1||Other arch will be removed|
|libavahi-common-dev||0.6.31-5||0.6.32~rc+dfsg-1||Other arch will be removed|
|libcairo2-dev||1.14.0-2.1||1.14.6-1||Other arch will be removed||#689122||✔||Also has deps on libfontconfig1-dev, libfreetype6-dev, & libglib2.0-dev|
|libfreetype6-dev||2.6.3-3.2||2.8-0.2||Other arch will be removed||#642354||See also #870618|
|libfontconfig1-dev||2.11.0-6.3||2.11.0-6.3||Other arch will be removed||✔||Partially||Marked as Multi-Arch: same; conflict due to dep on libfreetype6-dev|
|libgdk-pixbuf2.0-dev||2.31.1-2+deb8u4||2.32.3-1.2||Other arch will be removed||#689125||Reply to report implies gobject-introspection had priority... but see report #755874 for recent resolution; also needs libglib2.0-dev resolved|
|libgirepository1.0-dev||1.42.0-2.2||1.46.0-4||Other arch will be removed||Again, see #755874; also needs multiarch in libglib2.0-dev to resolve|
|libglib2.0-dev||2.42.1-1||2.46.2-3||Other arch will be removed||#683593||✔||Also see merged bug report #648621|
|libglu1-mesa-dev||9.0.0-2||9.0.0-2.1||Other arch will be removed||✔||✔||N/A|
|libgnutls28-dev||3.3.8-6+deb8u3||3.4.9-2||Other arch will be removed||✔||✔||Package already has Multi-Arch: same; conflict a side-effect of deps on libp11-kit-dev, libtasn1-6-dev, & nettle-dev|
|libgstreamer1.0-dev||1.4.4-2||1.6.3-1||Other arch will be removed||Will also need multiarch in libglib2.0-dev & gir1.2-gstreamer-1.0 to resolve conflict|
|libgstreamer-plugins-base1.0-dev||1.12.2-1||1.12.2-1||Other arch will be removed||Will also need multiarch in libgstreamer1.0-dev, libglib2.0-dev, libxml2-dev and gir1.2-gst-plugins-base-1.0 to resolve the conflict.|
|libgtk2.0-dev||2.24.25-3||2.24.29-1||Other arch will be removed||#689097||Partially||Several dependencies need multiarch to resolve conflict|
|libp11-kit-dev||0.20.7-1||0.23.2-3||Other arch will be removed||✔||✔||N/A|
|libpango1.0-dev||1.36.8-3||1.38.1-1||Other arch will be removed||#683592||See report #763851 for blocking files|
|libpulse-dev||10.0-1+deb9u1||10.0-2||Other arch will be removed||N/A||N/A||Partially||Already marked Multi-Arch: same; needs multiarch in libglib2.0-dev to resolve|
|libtasn1-6-dev||4.2-3+deb8u1||4.7-3||Other arch will be removed||N/A||N/A||✔||N/A|
|libxcomposite-dev||1:0.4.4-1||1:0.4.4-1||Other arch will be removed||#689082||✔|
|libxft-dev||2.3.2-1||2.3.2-1||Other arch will be removed||Will also need multiarch in libfreetype6-dev to resolve|
|libxi-dev||2:1.7.4-1+b2||2:1.7.6-1||Other arch will be removed||✔|
|libxml-parser-perl||2.44-2||2.44-2||Requires dependencies||Requires perl and perl-base to resolve.|
|libxml2-dev||2.9.4+dfsg1-2.2||2.9.4+dfsg1-3||Other arch will be removed||Requires libicu-dev to resolve.|
|libxslt1-dev||1.1.29-2.1||1.1.29-2.1||Other arch will be removed||Already marked Multi-Arch: same; requires libxml2-dev to resolve.|
|nettle-dev||2.7.1-5||3.2-1||Other arch will be removed||✔||✔||N/A|
|unixodbc-dev||2.3.1-3||2.3.1-4||Other arch will be removed|
I determined these packages by simulating apt-get build-dep with i386 build-dependencies for the (Debian unstable) version of wine-development:
apt-get -s build-dep wine-development/unstable -a i386
At first, this would return an error for some package; to bypass the error, I used apt-get source to grab the package source, quickly entered a "Multi-Arch: foreign" key into the control file, then rebuilt and installed it. After repeating this for a few packages, the simulated apt-get completed. I didn't need these packages to actually work, only to install and allow apt-get to give a full build-dep list.
Some of these pages might be out of date, but they're still good places to start: