I think you can achieve the desired effect like so:
Create a voice.
Make sure it is not set to playing.
Create a mixer.
Create and/or load all the streams you want to play simultaneously.
Attach all of those streams to the mixer.
Attach the mixer to the voice.
Set the voice to playing.
After you do this, you can manipulate each stream however you see fit. For instance, if you wanted to turn off a layer of music, you can set that streams volume to 0.
I haven't actually tried it, but will be doing so soon. I have a project which needs this exact functionality.
Append: Just tried this method and it seems to work correctly. I would advise configuring your mixer to use ALLEGRO_AUDIO_DEPTH_FLOAT32 to get the best results. I had a lot of clipping when trying to use ALLEGRO_AUDIO_DEPTH_INT*.
Append 2: This method might be overkill if you don't need exact synchronization. Allegro allows you to create as many streams as you want and attach them to whatever mixer you are using.