Possible, but not stable. It's a temporary thing that can last a couple million years at best.
In short, planetary rings may happen by two means:
1) a moon appears under the Roche limit altitude, and is torn apart by tidal forces. That forms a stable planetary ring.
2) some other means demolish a pre-existing moon; say, an asteroid that arrived from outside the galaxy and moves at speeds approaching relativistic. It would break up the moon/planetoid and form a ring for a time, but the ring will eventually coalesce into a new moon again.
Since case 1 is out of question, because the planets don't tear each other apart with their own tidal forces, you're stuck with a very improbable and temporary - but not entirely impossible option 2.
Unlikely additionally due to instability of the original orbit. The more spherical the central body, the more stable the moon's orbit. If there is no central body, any moon will be quite "temporary" as well - a captured asteroid will linger for some time, but the gravitational anomaly will either eject it again or make it crash into one of the planets. So not only did your binary planet system acquire some new moon recently, that moon had a bad misfortune of being hit by an object of a class so rare it only exists theoretically for now.
Also, I hope you're not planning for any life to have developed on these planets. The unstable gravitational field would make them quite an asteroid magnet, and the recent cataclysm resulted in a metheoritic rain so hard the whole surface turned to magma, and it's only solidifying currently.