What the call usually is
It starts with a request you would normally take. A booking system that has to talk to the client's accounting package. A portal their people log into. Something with data behind it, rules that change by customer, and a support life after launch.
You can subcontract it to whoever is free that month and carry the risk yourself. You can tell the client it is out of scope and watch them go looking. Neither is a good week.
How the work is contracted
You keep your client. We contract directly for the software work only: the build, the tests, the documentation, and the care afterward if they want it. We never sell into your scope, and we never take over the account. Design, strategy, content, the site itself, whatever you already do for them stays yours.
When it ships, you are the one who solved it. The code and the tests go to your client's own repository, and nobody had to be handed over to get there.
How the introduction works
Work that reaches us through you pays an introduction fee. What it comes to is set in the conversation, against the situation and the size of the thing, because a one-off build and an ongoing arrangement are not the same deal. We'd rather agree it with you than post a figure that turns out to be wrong for your client.
This already works
One agency has worked with us on exactly these terms for years. Their client list is still theirs, we have never quoted anything into their scope, and every conversation we have had with their clients has been about the software.
We don't name them, for the same reason we wouldn't name you: whose client it is stays their business.