Basement shower drain backs up with black muck whenever the toilet is flushed. What can I do to fix the problem?

You have a blockage downstream from the shower that must be snaked, You can snake nfrom the shower dfrain or if there's a bathroom group you can snake from thew lavatory roof vent. If you snake from 5the roof put out enough snake to reach the base and 15 foot more. Good luck, Tom