I'd glanced at that post while looking for a solution. It describes more-or-less what I've done; the problem is that when I add the second latitude measure the map context disappears: for some reason the map background no longer appears, aspect ratios go wonky, etc. So while the post was a good primer, it doesn't solve my problem.
So I've figured out the reason for this phenomenon, though I'm still working on a good solution.
I'd created two different calculated latitude measures based on the level of geography (PUMA vs CBSA); both returned Null for rows of the "wrong" geography type and the latitude for the "correct" type (e.g., [PUMA latitude] = null if geotype is "CBSA" and [Latitude] if geotype is "PUMA"). Then I dragged the two latitude measures to the Rows shelf.
It was the two different latitude fields that was the problem. Apparently only when you have two instances of the same measure does Tableau give you a map background. If the measures are different (even if they're both defined as serving the Latitude geographic role) Tableau treats the viz as a general Cartesian grid rather than as a map.
That's a bit out of my wheelhouse but let me know if you have any success resolving the issue. I'm sure others would be curious to know the result.
Could you change the layout of the underlying data so you only had one latitude and one longitude column, and make a variable which tells you which level of geography it is PUMA or CBSA?
Create your map as normal
Drag this new column into colour and right click on one of them (eg CBSA) and hide it from the map.
Right click on the latitude column to make a dual axis as you would normally
Duplicate your new column and drag the duplicate onto the colour of the new map.
Hide the other colour (PUMA), this should leave you with one map of CBSA and one map of PUMA then you can synchronize axes without losing the background map.
That's my normal technique for getting points on top of my custom made boundary maps.