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So, I currently have a bit of a hard time understanding the detachment rules and might just be blind to some obvious lines. Therefore excuse me should this be a dumb question that is clarified in bolt letters in the core rules.
I have a rather large collection, mostly of "stuff I like the models of/I have converted", but I would prefer to have it - at least in theory - organized in a way that would be legal to play. The vast majority is just Imperial Guard, sorted into a Brigade and an almost Brigade. But I have too many Elites and there are some models I have/want to get along the way from other imperial codizes and I want to build some IG Miasman redcowls with pure flamers that would best be represented by the Inquisition Acolyte datasheet.

Now I have glanced over the Inquisition rules and as far as I see there are no regimental traits etc. but just the Ordos. So if I, beside my two Brigades have lets say a Vanguard detachment with:
HQ: Inquisitor and SoB Canoness
Troops: AdMech Rangers
Elites:
Acolytes
2 x IG Bullgryn
SoS Vigilators
SoB Zephyrim
Fast Attack:
AdMech Sydonian Dragoon
Heavy Support: maybe 2 Kastelan Robots

1. would that be legal, as all have the keyword IMPERIUM and the slots for a Vanguard are filled?
2. the Admech don't get a Forgeworld bonus, the SoB no order Bonus, right?
3. would the Acolytes and the Inquisitor get their Ordo Bonus?
4. Could such a detachment use Inquisition Strategems/WLT/relics? (I assume no.)
5. Would including that keep the two normal IG detachments from getting their regimental boni?

Thanks,
Pyro

1. No, as you cannot use Imperium as the only shared faction keyword.
2.Right
3.Yes
4 Yes
5. No

Note answers 2-5 only work if not playing Matched play, as the detachment you showed is otherwise illegal due to the mixing of units.

For matched play, each 'army' needs to be in it's own detachment. You cannot mix Inquisition, Astra Millitarum, And Admech in the same detachment, as they don't share a faction keyword besides IMPERIUM. You'd need a seperate detachment of each.

If you want an inquisitor and retinue, they need to be their own Inquisition Detachment.

However, Psychic Awakening Pariah has rules for fitting a single Inquisitor into any Patrol, Battalion, or Brigade without breaking Faction Keyword rules. In this case, you can only take a Single Inquisitor (of any kind), and get access to their Strats and etc.

OK, that was fast and helpful.
Am I correct that the only way to legally include Acolytes then is within a pure Inquisition detachment?

 Pyroalchi wrote:
OK, that was fast and helpful.
Am I correct that the only way to legally include Acolytes then is within a pure Inquisition detachment?

Correct. Which is pretty lame, for anyone who wants to run Inquisition as a primary force.