I am a complete novice in this area but would love to reconnect my Tunze streams back to the apex and for them to switch to the batteries if the power or Apex unit dies. If you do anything in this area, I would really be interested.I admit I haven't fully thought out the idea yet and I have absolutely no experience with Tunze or Apex. I'm making some rather large assumptions here that I'd poke at a bit more if I go this route.
I'm not sure what happens if the Tunze doesn't get a 0-10V control signal. If 0 volts = "off," then a potential failure mechanism exists where the Apex 0-10 Vdc control signal output drops out but because the mains power is still present, there's no indication that it's inoperable. In which case, you'd need to monitor that 0-10V signal. You'd have to define some threshold under which you consider "failure" -- maybe 0.1 Vdc, or 0.5 Vdc, the level doesn't really matter for the point of discussion.
You'd use hardware (would probably need a bit more than a relay) to make sure that the control signal > your threshold. If not, you'd use the relay to kill the mains power coming from the Apex power strip, which would kick the Tunze over to battery power. Or if that doesn't default the Tunze to local control, you could use the relay to apply a signal from another source, which could be something as simple as a 5v wall wart.
There's a few ways you could go about it, really.