This is what I've determined.
The Righteous Fury rule states If any die rolled results in a natural 10 on a 1d10 (or a roll of total 10 on a 1d5) there is a chance for RF. Then if the hit results in damage, you roll a 1d5 and consults the relevant critical hit table.
My interpretation: This is not actual critical damage, just the effects are triggered (supported by the fact it isn't reduced by talents and traits, and it doesn't stack with normal damage or other RF hits). A target may suffer from multiple instances of RF, but they do not stack.
Now the tearing rule says that you roll an additional die for damage, and discard the lowest. Neither rule states clearly whether this additional die can trigger RF.
Then there's the additional problem of any weapon that deals multiple d10s of damage.
Options:
A single attack can only trigger RF once. - This makes the most sense to me, since the critical effect is special and not really associated with damage. This lessens the RF strength of tearing weapons and weapons that deal multiple d10's of damage.
Multiple instances of RF triggers off of one attack cause an additional 1d5 to be rolled, taking the highest result of the d5's for the critical effect - This is middle ground. This makes multiple d10 weapons and tearing weapons more powerful in RF than others, but really no different than if you roll really well for RF. It keeps the "cap" on damage from RF consistent with option 1 (a single hit will still only result in a max of effect 5 on the critical effect chart, and only one of them). Another option is multiple rolls and allow the attacker to determine which effect.
Multiple instances of RF triggers off of one attack cause additional RF effects. - This is the extreme. It makes tearing weapons and multiple d10 weapons much more powerful potentially.