In my household, it was understood from the start that the first wife (L.) would be the senior wife, and my second wife (S.) does defer to her in most ways regarding the running of the household. Not that there is always peace and harmony in our house.
An example:
When S. first moved in with us, she had two dogs, and we had two dogs. L. liked walking her dogs EVERY morning, and S. thought that was a good thing to do. Unfortunately for S., L. never changes her routine, so now that the two of them walk the dogs in the neighborhood, they will do it every morning that S. doesn't work. S. complained to me about it, and I had to tell her that if she doesn't want to commit to a routine, don't start one with L.
Now every morning that S. isn't here, L. walks a pair of dogs around the "loop" each morning, and returns for the other two dogs. On mornings that S. is here, the dogs get a real treat in that they get two laps around the loop. S. isn’t quite so happy about it, but she grumbles and goes.
I think the worst moment of friction occurred when we moved into our new and bigger house. S. is 4' 10". L. is 5' 7". S. can't reach any shelf above the first one in upper cabinets and likes to put her food items in the base cabinets. L. doesn’t want food in the base cabinets because it attracts critters.
To resolve the impasse, I had to build a “pantry” for food storage, and insist that all dishes go in the bottom shelf of the upper cabinets, and cookware go in the base cabinets. The upper shelves are only for my things and long term storage. I am glad they allowed me to rule by fiat on that one. I would hate for my house to fall apart over kitchen cabinets!
One technique that seems to mollify both sides is to talk to them separately, and ask for ideas to resolve the conflict. That gets the two of them in a more cooperative mood, and they settle their differences without me having to rule with a heavy hand. I am not really “Patriarch” material, and prefer to run my house by consensus when I can.
When I don’t rule in my house, it is grace, when I do rule, it’s LAW!