I would think the taillights would come on at the same time the headlights come on - for safety reasons.
Daytime running lights and automatic lights are two different systems. A car can have DRL and manual headlights, automatic headlights but no DRL, both DRL and automatic lights or manual lights with no DRL.
Daytime running lights turn the low beams on at 17-23% power or the parking lamps on every time the car is started and the parking brake is off. The tail lamps aren't turned on, nor are the rear parking lamps or interior night time illumination.
Automatic lights come on at dusk and turn themselves off at dawn (of course providing the car is running). Automatic lights turn the headlamps on at full power, the tail lamps and parking lamps on, as well as the interior night time illumination.
9th generation Corollas (American) have both DRL and automatic lamps. When we start our cars the DRLs come on, and when it gets dark our lamps switch to full power automatically.
I personally think this is the only way to go if a manufacturer decides to put DRLs on a car. If a car only has DRL and no automatic lamps, it is all to easy to look in front of you at night and mistake the DRL illumination for having your headlights on. If you do that, you are driving with weak low beams and no tail lights. That is dangerous, and sadly it is a common occurrence. I see it all the time, and I've almost hit several cars at night because they had only their DRLs on and had not turned on their full lighting system.
There should really only be two types of systems. DRLs with automatic lamps and manual lamps with no DRL. That way people with DRL won't have to worry about turning on their lights and people with manual lights won't have DRL to fool them into thinking they have their lights on.