If the corp is owned by the gov't then it can't be owned by the workers. The equity is the ownership. What could be done is the gov't could be extending loans or provide grants, plus act as a customer that pays for the service the coop provides. For example the gov't wants to build low rise units, it helps establish this coop construction firm and gives it a contract to build units in Montreal. The coop takes that and fullfils the project.
With that said, this may not be in the rest of our interest as the firm might start prioritizing other private contracts that its workers deem more profitable. This is why, the gov't has to maintain some form of ownership, whether full or some controlling share scheme line PRC-style golden shares.