I found an iOS app that can play audio while other audio is playing. It's called EverMusic, and it's pretty accessible and feature-complete. It can play local files, connect to network devices and cloud services, and play internet streams. It also has an equalizer that includes a preamplifier, so if you want to play a radio stream at -24 DB while your audiobook plays at normal volume, this is the app for you.It doesn't do this simultaneous playback by default, mind you. You need to go to Settings, Audio Playback, and change the "Audio Output Mode" to "Mixed". At that point, you lose the ability to control playback with a headset or from the lock screen, but if you start audio playback from another app, your headset and lock screen controls will apply to that app, and EverMusic will keep playing in the background no matter what you do.There are also at least two versions of the app with slightly different feature sets, so if you like the app and you want to use it for normal playback as well, you can always just use EverMusic Pro for that, and keep the free EverMusic for your background playback. I haven't tested this, but it might be possible to run both of them at once.Coming from Windows where you can casually load up 12 media players and send them all to different audio devices, it feels strange to be excited about an app developer taking the time to thoughtfully work around an audio stack limitation, but I'll take the win and give the developer my thanks.