In this new edition we have new ways to build armies. Ignoring unbound for now as there is little to discuss there (bring what you want). In the rule book there are three official detachments, Combined Arms Detachment, Allied Detachment, and Formations. These detachments have certain rules and restrictions that change how armies are built from 7th.
The big change is the addition of factions. On page 119 it describes factions and has a handy chart to show the different ones. In addition it explains how certain things interact with each other. The big one is codex supplements, for all intents and purposes they are the same faction as the codex they supplement, ie black legion is the same faction as CSM. On 122 it describes how allies and there is an important restriction, that being that it cannot be the same faction as your primary detachment. So no longer can you ally black legion with CSM/crimson slaughter.
There is another important thing to note, primary is not a type of detachment, just the main force where your warlord must come from. Currently only CADs in battle forged can be a primary detachment. Now when building multiple CAD's you must choose units from the same FACTION, not codex. As written this means that while you cannot ally with yourself, it is completely legal to take one CAD black legion and another CAD CSM. I know there is a lot of butt hurt over the allies thing but this is in many ways better than allies as you no longer have restrictions on building your allied force.
There is some interesting wording of how you build each individual CAD in how it mentions factions, and can be interpreted to mean that even a single CAD can take units from multiple codex/supplements as long as they are of the same faction. Now I think this is silly, but it could be a valid way to look at it so as much as I think you should stick to a single codex, nowhere is it mentioned that models must be from a single codex, Faction is term used a lot in this section. For most armies faction and codex are one and the same, but as time goes on and more supplements come out this will be less of the case. Hopefully an FAQ will be included in the main BRB to clarify that while you can have multiple CADs from different codex's of the same faction, a single CAD must use a single codex. In any case this is an easy work around as we as a community can simply play it as we see it to mean. If your worried about organized play, have no fear, most tournaments have already made it clear that a single CAD is one codex.
Now there is some mention of no faction, this could mean supplements that have no real faction, the only one I can really think of is cypher, though this could apply to Militarum Tempestus. This is where it comes back to the IG player. Tempestus is not a supplement and therefore does not qualify for the rule on 119 as supplements having the same faction as parent codex. It is it's own standalone codex, though it is an army of the imperium for allies purposes, the quest does arise whether IG and Tempestus can ally. Its an interesting question and until an FAQ covers it there is no solid answer. If it is considered to be the same faction as IG then you are still fine and can take a second CAD for Tempestus. If not then you can ally in and use it that way. In either case, tempestus and IG can play together.
Well I decided to bust out the tempestus codex while writing this (crazy right?!) and see if there was any clarification in it and in a way there is. It is described as being its own faction (not Astra Militarum) and battle brothers with all armies of the imperium, it has the same chart as armies of the imperium, same desperate allies, same come the apocalypse, and convenience. Its faction is Militarum Tempestus, with same allies rules as Armies of the Imperium. Rereading the section it is clear that it was written with 7th edition in mind. So there is your answer, Tempestus is a different faction from Astra Militarum and can ally battle bros with all Armies of the Imperium. Hope this helps.