Not quite what you're looking for, but the public service, local councils, state owned companies etc often work this way internally.
Eg local council has a council as such (ie elected members) with administration etc. And it has a works unit that does roadworks etc. Everything done by the works unit is charged back to the council proper in the same manner as if it were done by a contractor. And some of the works undertaken by the council proper aren't done by its' own works unit - in some cases the works unit has to tender for the work.
Same in some cases in the public service - state govt ends up effectively being a "contractor" to the Australian Government in many cases.
Same with state-owned companies. Eg Hydro Tasmania generates electricity in Tasmania. It has a subsidiary, Roaring 40's, which operates power generation in Tas and SA. Plus there's Entura which is an engineering consulting business operating within Australia and internationally. And then there's Momentum which retails electricity in Victoria - power that they simply buy from rival generators in order to retail it in Vic. All are owned by the Hydro-Electric Corporation, itself owned by the Tas state government. They all have the same logo, although Entura and Momentum each use it in different colours.
No doubt there's quite a few similar situations in the private sector I would assume. I don't recall names, but I've often dealt with companies that are part of a "group" in this manner.
I'm not certain, but I suspect the likes of Wesfarmers may operate this way in practice - Bunnings is one part, Coles is another, producing LP Gas is a very different business, with only a minor amount of that gas sold by Bunnings, but ultimately it's still part of Wesfarmers.
Gunns is primarily a forestry company but until very recently also ran hardware stores. Presumably the timber sold at the hardware stores came from Gunns' own sawmills, but apart from that it would seem to have been a separate business.
Telstra is another. A network owner/operator and also a telecommunications retailer which has just started offering services via networks not owned by Telstra. And of course the network side of the business also sells network access to other retailers who operate in competition with Telstra in terms of retailing telecommunications.