Loyalty status is the final stage in the relationship between yourself and your Underboss.
Once you reach Loyalty staus with one underboss you unlock this achievement (Trust). However gaining Loyalty status with ALL three under bosses will unlock the family achievement.
There are two methods of working towards this
Method 1You must assisgn a MINIMUM of three Districts to an Underboss. Once they have been assigned you must then begin there Side Miissions. for Vito I Need A Favour, for Cassandra .45 In My Hand & For Burke, The IRA Dont Ask. You then need to continue the side mission until you reach the final stage.
This will unlock the achievement. You will need to do the above method if you want the family achievement.
Method 2 Once you assign the 6th district to ONE underboss then the achievement will pop. You will not be required to do the side mission however you will not be able to gain the Family achievement.
The racket building side missions are not a requirement
I unlocked this achievement with Burke I had assigned 3 Districts and was at the last stage of his side quest when this popped.