I'll give it a go...
Definitions:
Disallowed - Simply not allowed, reserved purely for same-fleet allies.
Trusted - People who might get along by choice when pressed
Necessity - Either given their marching orders, or under the understanding that they'll get back to killing each other afterwards
Distrust - Ongoing concern about "friendly" fire, suspicious "weapon systems failures" and conspicuous absence in battle
Hostile - Shoot first, shoot some more; don't leave anyone alive to ask questions.
Disallowed:
You cannot take allies from your own type, ie IN/IN, Chaos/Chaos etc
Trusted zone:
Admech: Imperial Navy, Space Marines
Chaos: Ork, Eldar
Dark Eldar: Eldar, Rogue Trader
Eldar: Dark Eldar, Rogue Trader
Inquisition: Imperial Navy
Imperial Navy: Space Marines, Admech
Necrons: Admech, Chaos
Orks: Chaos, Imperial Navy
Rogue Trader: Inquisition
Space Marine: Admech
Tau: Rogue Trader
Tyranids: Necrons
Allies by necessity:
Admech: Imperial Navy, Space Marines, Inquisition
Chaos: Ork, Eldar, Tau,
Dark Eldar: Eldar, Rogue Trader
Eldar: Dark Eldar, Rogue Trader, Chaos
Inquisition: Imperial Navy, Space Marines, Admech
Imperial Navy: Space Marines, Admech, Inquisition, Rogue Trader
Necrons: Admech, Chaos, Inquisition
Orks: Chaos, Imperial Navy, Rogue Trader
Rogue Trader: Inquisition, Imperial Navy
Space Marine: Admech, Imperial Navy, Inquisiton
Tau: Rogue Trader, Eldar
Tyranids: Necrons, Chaos
Distrusted:
Admech: Imperial Navy, Space Marines, Inquisition, Rogue Trader, Tau
Chaos: Ork, Eldar, Tau, Imperial Navy
Dark Eldar: Eldar, Rogue Trader, Chaos, Tau
Eldar: Dark Eldar, Rogue Trader, Chaos, Tau
Inquisition: Imperial Navy, Space Marines, Admech, Rogue Trader
Imperial Navy: Space Marines, Admech, Inquisition, Rogue Trader, Chaos
Necrons: Admech, Chaos, Inquisition, Imperial Navy
Orks: Chaos, Imperial Navy, Rogue Trader, Space Marines
Rogue Trader: Inquisition, Imperial Navy, Space Marines
Space Marine: Admech, Imperial Navy, Inquisiton, Rogue Trader
Tau: Rogue Trader, Eldar, Imperial Navy
Tyranids: Necrons, Chaos, Orks
Anything left is hostile.
If anyone wants to throw that into a chart like the previous one, that'd be amazing as it would give much better oversight over who has what allies and if anyone has too few or too many.
This list has been compiled mostly thematically. Some of the allies are only one-way, representing people you're willing to assist because you can stab them in the back later, but you'd not want them covering your rear, or largely where the Imperium is concerned, you've been ordered to assist them but would not trust them yourself.
Tyranids have gained some friends, though they only just about trust the Necrons because they've figured out they're not very tasty... However, nobody trusts them, so nobody gets them as allies.
Some people (especially Orks and IN) have allies through respect earned by fighting each other (a bit like Commissar Yarrick).
Obviously open to suggestions, and I'd be happy for a table to be based off this lot. The trusted allies were largely the core of the idea (some of them were made up on the fly), but most of the other grades of ally were made up on the fly.