I have noticed that on all the tandems I have seen the pedals are mounted in the same position. i.e. while the front right one is up, so is the rear right one, and not front right up and rear right down.
Is there a specific reason for this?
Having the pedals 180° out of sync would lead to a couple of problems:
A phase difference of 90° is used sometimes. The power is more even through the stroke but handling is worse at low speeds among other downsides.
The second article suggests that most people ride exactly in phase because that's how tandems are set up by default. Changing takes some getting used to and a little fiddling so people need a good reason to mess with it.
My wife and I have ridden a tandem for 2 years. Shortly after buying it we wanted to ride High Pass Challenge, a ride up to an observation point on Mt. St. Helens. As newbies, we were not very good at stand-up pedaling together and I had to keep a death grip on the handlebars to keep the tandem going straight with the pedals in synch and my wife's weight shifting in synch with mine. We did that HPC ride with the pedals 90 degrees out of phase and it made my job much easier with a very smooth power input to the bike. However, we quickly found out after the ride we wanted to get back in synch for normal flat riding and here is the key. With pedals in synch, we push the pedals together and my wife immediately senses when I back off my power. When we are out of synch, her power strokes are always on her own and when I back off it takes a couple of crank revolutions for her to figure it out. It feels like a 4 cyclinder engine running on 2 and she really hates it.
So in short, I prefer out of phase but my wife prefers in-phase and to avoid divorce, we keep the tandem in phase for her. We are much better tandem riders now and do all our rides, even mountain climbs, in synch. To climb in synch the captain just needs to train the stoker to keep the weight centered at all times and warn of any unexpected weight shifts (water bottle, stand-up butt breaks, etc.).
As others have pointed out, you can't put the pedals 180 degrees out of phase without having the stoker's pedal hit the pavement when cornering. 90 degrees puts her level while the captain goes down on the outside, or the other way around if the captain is confident enough to keep track of which of the stoker's pedals will be down in a corner, but that seems like a recipe for a disaster that will likely end the tandem experience if the captain gets it wrong even one time.