I messed with this for hours and could not come up with a solution.
I read on some Android-x86 forums and found out that you can open a terminal emulator and mess around with some alsa commands. The only problem is they won't work unless you are root which means they are useless until we can get root in Remix OS.
Some notes:
1. We can't patch this in obviously, but the 'ordering' made me curious to see if it is just choosing the wrong output.
2. So I ran 'cat /proc/asound/cards' and saw the Intel HDA showing up just fine, but I didn't see anything major relating to HDMI, but the Intel one was set as card 0 so I didn't see any major problems there.
3. Like I said, without root access 'alsa_aplay -l' will always just output "No sound cards found" which is annoying.
4. You also can't edit or add any files in /etc because no root access.
At this point I hit a wall. One idea I had was that we can try adding in a custom alsa settings file by editing the data for /etc on the USB drive outside of Remix OS. This would basically bypass root access in theory.
Well hopefully this brain dump gives some other people some ideas. Good luck!