True i386 support would mean compatible with the original Intel 386 processor from 1985. The 486 added a few additional instructions in 1989 but things really changed with the Pentium in 1993 - that gave us i586 which is the bare minimum for most modern software today. Much software can still run on regular Pentiums today if compiled for it, but SSE2 optimizations requires at least a Pentium 4 or Core CPUs instead.
I play with retro PCs often and found OpenBSD's i386 target stopped supporting real 386 CPUs after the 4.1 release, and dropped support for i486 somewhat recently in 6.8. It now requires at least a Pentium class CPU, i586, though the arch is still referred to as i386 likely because it's a common proxy for "32-bit".