The best opposite of "mutually exclusive" I can think of is "necessarily accompanying", but it sounds awkward.
Most answers I looked up give words like "concordant" and "accompanying", but these words have more passive definitions that mean things are "compatible", "harmonious" or "in agreement".
I want a word with a harder definition that means things must exist together or not at all.
Best Answer
In philosophy the expression would be mutually necessary:
Example of mutually necessary conditions:
In a broader context:
Interdependent implies a certain level of mutual necessity:
Codependent relationships exhibit an intensity of interdependence that increases the sense of necessity:
Symbiotic relationships do not all exibit necessary, but the mutual benefit is a specialized expression of interdependence:
There is a brand of symbiosis referred to as obligate symbiosis, which does imply the must of the OP:
Synergetic, a derivative of synergy approaches the meaning you are looking for: