It looks like Arch does not have mingw64 cross-compilers in core package
repository. It does have some package in the user repository (AUR), but
I assume that such a repository cannot be deemed as safe.

Anyway I still tried, but apparently these AUR packages have to be built
and when I tried, I got this error:
>  ERROR: Running makepkg as root is not allowed as it can cause
> permanent, catastrophic damage to your system.

Anyway it's all a big mess. Then I tried to move the cross-CI to Debian
testing, which is anyway our base compatibility system. Unfortunately I
encountered like what looked like some glibc++ macro problem on some
packages (most likely because the pre-built packages I use are Fedora
ones which likely uses a cross-compiler differently built from the
Debian one).

So in the end, for simplicity, I use a Fedora image, then I am sure to
get a good match between the system cross-compiler and the pre-built
dependencies.