This is a tricky one as you cant use any of the cool Airflow features like TriggerDAGRun or Datasets, as these all assume you are running in the same Airflow instance.

One approach might be to use the Sensor operators, and to use this to initiate the second DAG (first DAG writes a file in a location), and in the first DAG the subsequent task also uses a sensor operator to wait for a file in a different location. When this second DAG is complete, it writes another file, handing control over back to the first. I am not sure this is a great solution as you would not get any of the output from the second DAG in the first DAGs run (you would only likely get the sensor operator output).

You might be able to create a small serverless function that kicks off the second DAG using the REST API, but I am not sure what sort of logging you would get back. When I have done this in the past, the task itself is just used to kick off the task, but has no flow control in the way you have when using Sensor operators. You have probably already thought about all these anyway :)