Just to clarify, in the MinGW case we are interested in:

"build" = Fedora Linux, usually i386 or x86-64 (but not always)
"host" = Windows i386

We can, to a limited extent, run the host binaries on the
build system, using Wine (the Windows emulator).  This doesn't
always work, and in any case is usually regarded as the wrong
thing to do because Wine is a very large and complicated build
dependency which requires, amongst other things, a working X
server.  Also since Wine doesn't run on anything which is
not x86-like, if we have to run Wine during the build then we
cannot cross-compile from other Fedora build systems, specifically
ppc and sparc.