From what I can tell, when characters begin an argument it follows this pattern, so long as it is not broken:
- Characters have automatic lines that spark an argument.
- The argument results in a dialogue-box conversation. [Where typically the Watcher has some input on the fight.]
- More dialogue plays between the two characters.
- There is a dialogue-box resolution. [it is not always automatic, sometimes you need to talk to one of the pair and note that they "look like they have something to say."]
So for example: After some conversations, Tekehu places a small carved goddess charm in Pallegina's pack. Pallegina is outraged, and the two argue. After this fight Tekehu and Pallegina have a few more dialogue pieces. Then Tekehu crafts a small Vailian ship out of water, and presents it to Pallegina as a gift. Pallegina realizes that Tekehu is just trying to be friendly, and forgives him. She keeps the sculpture and there's no more dialogue between them. It is a very sweet scene.
I have not been able to get every companion combination to go through this arc, but at least half of the major arguments did. Some friendship paths also take the same pattern, such as Maia/Xoti. I did a lot of running back and forth between maps to trigger the automatic dialogue.
Tekehu will also comment on your current relationship if you have locked one in before he approaches you. I think his dialogue towards Aloth was something like, "Really, the academic?" He seemed surprised but not upset. Later on he talked to Aloth and said he was surprised Aloth did not run around telling everyone. Its very cute!