The Audiovox unit has two ignition connections---the thick blue and the thick green. If the blue Ignition 1 wire is seeing power, the alarm will assume the car is running...and you won't be allowed to arm the alarm. This MAY also be true of the green Ignition 2 wire, but it probably isn't the case...but you'll have to try it to find out.
So, you'll need to prevent the Audiovox's ignition 1 wire from seeing any power while the turbo timer is on. This should be pretty easy to accomplish with one relay, but I'm not familiar with the GReddy's wiring, so you may have to figure out some of it for yourself. Also, I will try to explain the setup in a way that is easy to understand, but may not make for the most elegant-looking install....after you've figured out the concept, think of a way to arrange things so it's neat and simple.
The Audiovox's blue wire will exit the alarm brain, and be connected to the car's ignition wire. Next, you'd cut the blue wire right in half----connect the two ends to terminals #30 and 87a of a relay---it doesn't matter which wire goes to which. Terminal #87 will be unused.
In its rest state, the blue wire will be "normal"---that is, it will behave as if the relay weren't even there.
When the relay is energized, the blue wire will be broken, and therefore the Audiovox unit will think the ignition has been turned off.
Terminals #85 and 86 of the relay are what makes it energize---one of them must see positive power; the other must see ground. Take a look at what outputs you have available from the GReddy unit, and figure out some way to energize the relay only when the turbo timer is on.
NOTE: You'll also have to remember that the shock sensor might trigger from the vibrating engine. You'll have to set the shock sensor low, or use yet another relay to interrupt the shock sensor when the turbo timer is on.
I like nighthawk2k's suggestion of using passive arming for the alarm, but there is kind of a catch to that. To make the alarm arm all by itself, you must turn off the car's ignition, and then open and close a door.......so not only does the blue ignition wire have to go dead....but the brown door trigger input must receive a pulse. If your turbo timer has an output that pulses AFTER the timer shuts down, you may be able to make use of it.
Another consideration, although it's a big compromise, is to just set the remote starter for a 5-minute runtime and forget your turbo timer altogether. When you arrive home, simply put the remote start into "pit-stop" mode, remove your key, exit the car, and arm the alarm....the engine will shut off in five minutes.