If a character with the Unarmoured Defence class feature casts Mage Armour, what is their AC? Do they choose which 'base' AC they take, or do they stack somehow? Is there rules text that explains this conflict?
They don't stack: the player chooses one.
Both of them set your base AC. So mage armor would supersede Unarmored Defense if it offered an increase.
You have this:
- Normal: AC = 10 + Dex
- Unarmored Defense: SET AC = 10 + Dex + Stat
- Mage Armor: SET AC = 13+Dex
Basically, mage armor leaves no room for the second stat for Unarmored Defense so you would not be able to apply the second stat.
The exclusive nature of AC calculations was explained in further detail in a post at the WoTC web site in 2016
These methods—along with any others that give you a formula for calculating your AC—are mutually exclusive; you can benefit from only one at a time. If you have access to more than one, you pick which one to use.
There are two ways to alter your armor class:
Set your base AC
Add a modifier to AC.
You only get the best base AC available to you, and not many things add a modifier. Mage Armor sets your base AC and Unarmed Defense from the Barbarian and Monk class features set your base AC as well, thus you only get the base AC from the best of them that you qualify for. But they are the same thing despite being worded slightly differently, per this Twitter dev response:
"Is there a reason why the wording is inconsistent between Mage Armor and Unarmored Defenses? Mage Armor uses "Base AC". The difference isn't intentional. -J"
There are very few modifiers for Armor Class (and this is intentional according to WOTC), but notable ones are Shields, Defensive Fighting Style, and the Dual Wield feat.
This question has been answered by Jeremy Crawford in Sage Advice:
Does Unarmored Defense work with a spell like mage armor? Unarmored Defense doesn’t work with mage armor. You might be asking yourself, “Why don’t they work together? Mage armor specifies that it works on a creature who isn’t wearing armor.” It’s true that the target of mage armor must be unarmored, but mage armor gives you a new way to calculate your AC (13 + your Dexterity modifier) and is therefore incompatible with Unarmored Defense or any other feature that provides an AC calculation.
Mage armor is a magical force with no weight to it. It is not real armor. Unarmored Defense for a monk means being able to freely move, so mage armor does not interfere with Unarmored Defense.
So to agree with the previous post, yes you are changing base AC once to 13 and adding your dexterity modifier as normal for mage armor, then adding your wisdom modifier for Unarmed Defense. Furthermore, if you use the dual wielder trait and are fighting with two weapons, you can get another +1 to AC.
Now as a DM if my player does this and I let him, he is very happy. Also it makes sense to us aesthetically. Now it is up to the DM to come up with ways to make encounters still fun and challenging. No need to be a rules lawyer; just have fun, especially with newer players.