All healthy relationships are conscious, intentional and thrive on differences! In order for you, your colleague (and the two teams you are leading) to creatively solve the “falling Net Promoter Score,” and other issues, you all must learn to disagree without being disagreeable! Innovation is forged in the fire of passion and hammered into shape on the anvil of contention!
Besides the two fine references mentioned about, consider studying Marshall B. Rosenberg’s book: “Nonviolent Communication.” http://amzn.com/1892005034 This is a simple process for reconciling (not just resolving) disagreement without hostility and long term damage (i.e. loss of face).
Our Authentic Leadership model supports healthy relationships with a Designed Alliance that safely meets the needs of both individuals and eliminates un-negotiated expectations. From the limited information you have presented, I make up that you and your colleague have some un-negotiated expectations of each other. One, or both of you, feel let down, disappointed and angered by the other’s actions (or in actions).
The longer this angst festers, un-reconciled, the more damage it will do to you and your team. Soon it will be covered over by “professional” behavior. No one will speak of the low-level chronic anger; it may even sink below consciousness. But it will continue to contaminate the relationship between the two teams.