Ok, most fish, especially those in lfs settings have already adapted to the same water conditions. It's easy to find a middle ground if you know your water parameters. We can loo kat the recommended water parameters for both types of fish and some commonly kept species. Others, I'm aware informations will vary from place ot pplace so please don't nit pick too much.
You can fidn the info on fishprofiles
The average stats for a Jack Dempsey
pH: 7.0 - 8.0
Temperature: 72F - 86F
Hardness: 10 - 20 dH
Carnivore
Average Stats for a Convict
pH: 7.0 - 8.0
Temperature: 68F - 79F
Hardness: 9 - 20 dH
Omnivore
Both of those are NW cichlids
Average stats for Peacock cichlid
pH: 7.5 - 8.5
Temperature: 73F - 81F
Hardness: 10 - 15 dH
Carnivore
Average stats for a Yellow Lab
pH: 7.5 - 8.6
Temperature: 73F - 82F
Hardness: 9 - 16 dH
Omnivore
Both of which are African Cichlids, more specifically Lake Malawi
So if you have the proper water, You're stats could be
pH: 7.5-8.0
Temperature: around 75F
Hardness: around 10 dH
Diet, I feed a vary varied diet, I also have bottom dwellers. I feed spirulina, various frozen foods, flake pellets etc.
Temperment, this is jsut knowing your fish. I watch the fish very carefully, sometiems for hours, and I watch the heirarchy in the tank. I always pick the fish at the bottom of the pyramid to ensure the most "tame" "docile" fish. I rearrange the tnak, all fo my tanks have caves, rocks and sand as a substrate, soem have driftwood/bogwood.
Its basically stretching beyond the "norms". Also, it was only a couple africans in a tank full of NW cichlids. They were all juvies and pretty much grew up together. I don't recommend that peopel do this, but it is possible. Right now I only own one african and he gets on fine with all of my other cichlids. He knows his place al lthe other fish know their place. My fish NEVER fight, well seriously fight anyways. Its feeding tiem you see this organization take place. Like I said, I only have one african left and he's been i nthe situation for a while.
Its possible, but I won't recommend it. Its all about knowing the disposition of all of your fish, knowing your water parameters, and being able to judge the aggression correctly. In cichlids I feel thsoe huge humps are ugly, I try ot avoid them, hence my fish aren'taggressive. I don't liek glass bangers.
I find fish that have been through a rough life and you look into their eyes and you can see it there. You take them home, they thank you. All of my fish were "rescued" fro ma life they didn't like, and honestly I feel in return they thank me by tolerating fish that they don't like
. Well all except my festae
. But the others get along great.
Hmm... hopefully I don't get reamed by other members for keepign the african in there, but what are you supposed to do when I wanted a fish for my JD and they recommended a fish that later you found out isnt compatible??? You make due. And he's about 6" full bodied, I swear he learned ot be a NW cichlid so much he bulked up liek one lol. You couldn't tell the difference between him (an albino zebra I think he is) and my pink con male if you looked quickly
Yuo had to look for the slightest difference in height and the "egg spots" o nthe african.
But I suppose there could be more....
And just to clarify, Central and south American Cichlids fall into New world Cichlids. Just making sure we dont confuse anyone 
