Well, weíve already answered a question of what happens when a positron and electron annihilate -- it depends quite a lot on whether they had some initial kinetic energy and how much. In your case, it seems as if you are assuming there is no initial kinetic energy. An electron and a positron in close proximity with very little kinetic energy in the system will often form a short-lived bound state called "positronium". The lowest energy states of positronium are called "orthopositronium" and "parapositronium" depending on the total spin of the electrons (pointing in the same or opposite directions -- actually linear combinations of these).
The spin-1 state never decays to just 2 photons -- it decays to three photons. In fact, there is a good chance of finding a third (or more) photons in the final state due to bremsstrahlung -- the lower in energy you look, the more photons you will find.
This has an effect on both the statement about "opposite directions" and "in phase". If there is a third photon, it may keep the two highest-energy photons from being emitted in opposite directions by conservation of momentum. Also, a tiny amount of total momentum of the e+ and the e- will make the emitted photons in general not go off in opposite directions.
Initial momentum and additional photons also spoil the "in phase" part of your question. If there is some (any!) initial momentum to the system, then the photons will receive different doppler shifts as the system is boosted back to the laboratory frame, and therefore will have slightly different frequencies, and therefore will not be in phase. Similarly for the emission of third or more photons -- in general that will make the other two have different energies.
If there are only two photons over a very low energy threshold (which will happen in a fraction of positronium decays), then I imagine that you can arrange a pair of mirrors and collimators which will bring the photons back together and you can test to see if thereís an interference pattern, and there probably will be a phase coherence between the photons over a short enough distance (let the photons propagate far enough, and this may even be a tiny distance! and tiny differences in their frequencies will spoil any initial phase coherence).
The polarization of the photons is also correlated -- the total angular momentum of the photon system has to add up to the total angular momentum of the positronium before it decayed, and so the polarizations will be correlated in a way that depends on whether you started with orthopositronium or parapositronium.
Let me try a somewhat different answer to what I think the core of your question is. Letís assume that the net momentum of the initial particles is zero- thatís just a choice of the center-of-mass reference frame. Also. letís assume that they have enough initial energy to avoid forming a bound state, and that exactly two photons come out. Then youíre right that the photons must head opposite directions to keep the momentum zero. (Actually, what nature know is that there are two photons heading opposite ways, not which particular ways they are heading.)
Now the question about phases is a little tricky. I think youíre asking whether the electric fields at some fixed distance from the collision and some fixed time after the collision in opposite directions (all in that same frame) point the same way or opposite ways. The first thing shocking thing to realize is that these quantum objects do not have sharply-defined particular values of the electric field. In fact, for a precisely defined number of photons the possible values of the electric field are distributed around an average value of exactly zero! The second shocking thing is that although nature does not know which way the fields point at some time and place, it does know whether the fields at two opposite points are the same or opposite. If you think of the fields when two classical oppositely charged particles approach a collision symmetrically, youíll see that at opposite points around the collision, the fields point the same way as each other. So I think that that symmetry is preserved in the collision. I guess that means the photons are what you call íin-phaseí, although they donít have separate phases.
On the polarization of the photons, I canít answer since you havenít specified anything about the initial spins of the particles.
Whew, tough question.
(republished on 07/20/06)