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[WH40k] What would it take for a "civil war" to spark between the Ministorum and Mechanicus in a sector?

Say a system in a sector contains both a Shrine World and a Forge World, separated by a rich asteroid belt that has causeda great amount of tension over mining rights between the two factions. Then, unknown opportunists causes something catastrophic to one of those two worlds, who naturally blame the other, and is seen as an act of war. Knowing that both these factions are utterly integral to the survival of the Imperium's survival, how would this conflict manage to escellate into a sector-wide conflict? And how long/big would the conflict need to be before the rest of the Imperium is forced to squish it?



Edit: While both the Ecclesiarchy and the Cult of Mars maintain their own fleets and military, I wonder how the Sector's Naval Battlefleet would react; as it is woefully unclear if one or the other is rebelling against the Imperium; would they simply attempt to blockade the warring fleets from eachother? Attack them both?
gpt
The Ministorum would need evidence of heresy in its Mechanicus neighbors to justify a war, and the Mechanicus would only be goaded to conflict by the destruction of a much-revered technological marvel. 

In the former case, the Inquisition would likely be involved before escalation to war, striking at the top of the Hierarchy and recommending further cleansing action thereafter.  In the latter, the Mechanicus would need a damn good pretext to avoid being blasted into oblivion by nervous or opportunistic fleet commanders; namely, that whatever the Ministorum has stolen or compromised was essential to some nearby war effort.  At that point, they rely upon leveraging the military hierarchy against the Ministorum rather than bringing the matter to war outright.

TL:DR  By the time the tension escalates to war, it'll be a matter for the Inquisition or Sector Generals.