Normally if you had 3 bits of text with thick strokes applied to each and then pushed them this close together you wouldn’t see the same thing illustrated above. The strokes of the ‘higher’ objects would overlap onto the lower objects. Then just how was this accomplished here?
Take all three text objects and Group them. Then Highlight the Group in the Appearance Panel. Next, add a New Stroke. The new stroke will appear on top of all other Appearances so Select the Stroke and drag it below the Contents. That’s it.
Now you can double-click the group to enter Isolation Mode and freely move the text objects around, and the group stroke will faithfully follow along. (As demonstrated below.)
There are some potential problems with this technique and you’ll need to avoid them. For example, if you have a Drop Shadow applied to one of the shapes inside the group, then your ‘group stroke’ will wrap around the outer feathering of the shadow and won’t look right at all. Another potential problem is if you already have a stroke applied to one of the objects inside the group. Live Strokes are… live, so they don’t get considered when adding a stroke to the entire group and will be ignored.