This story was a spin-off of the novel Egg Benedict, about Kyle (Karlyle's brother) and Nick (current husband, but not at the time of the novel). I haven't been able to find it in English :( But I did read a summary a while ago.
LIGHT SPOILERS
In it, Ash was actively courting Nick, which is why it's implied in Define the Relationship that Karlyle and Ash had hostility towards each other. Karlyle was mad at Ash for getting in the was of Kyle's relationship, and also bitter that Ash didn't recognize him from New Year's (when they first reunited, Karlyle lashes out that he's worst for this, which Ash alludes to in this comic).
In the epilogue(?) when Kyle and Nick are married, Nick calls Karlyle for a favor, which as you know is essentially to have sex with Karlyle. Kinda messed up if you ask me because:
1. Karlyle and Ash clearly didn't like each other (at the time)
2. Ash was deeply in love with Nick, so even if it meant sleeping with another, Ash would do anything he asked. IIRC, Ash even says "If you really want this and it'll make you happy, I'll do it."

I love this story, I think that Karlyle is a precious character, but I've always wondered: was there a previous story to this? In the first chapters they talk about a previous meeting that left a bad impression, but we are only shown the kissing one, does anyone know?