How do you have two Sims to have "awful compatibility" ?

I usually try to accomplish good compatibility by giving them matching likes for conversation topics and have them like sim characteristics that the other sim has. So I think you should mismatch conversation topics, have one sim like gossip and the other sim hate it. Or have them dislike each other's traits. For example if one sim has the  loves the out doors trait then give the other one a dislike for nature enthusiasts.

Quick test. I created two sims

Sim 1 sceptic, nerd and childish 

Sim 2 snob, high maintenance and art lover

Test 1

• Sim 1: dislike artsy sims
• Sim 2: dislike intellectual sim
• Result: bad compatibility 

Test 2

• Sim 1: dislike artsy sims 
  
  Sim 2: dislike intellectual sim
• Add opposites on conversation topics deep thoughts, small talk and stories 
• Result: bad compatibility 

Test 3

• I tried to add more dislikes and mismatched conversation topics. Even had one sim hate colours and music loved by the other sim but wasn't able to get awful compatibility in my quick test.

I have seen awful compatibility in my game but usually try to avoid it. Getting awful compatibility on purpose seems more difficult than I thought.

I think you best look at this list by EgonVM​  to get more information. It includes how much each trait weighs and you need to get to - 6 or lower for awful compatibility. 

https://forums.ea.com/discussions/the-sims-4-general-discussion-en/compatibility--self-discovery-explained/5002975/replies/5003198