Reloading string table assets breaks references to strings and logs warnings

When using the "Reload" asset action on a String Table asset, any Text variables that reference the string table would break and say "<MISSING STRING TABLE ENTRY>".

(From licensee)

Steps to repro:

Create a string table
Add a string
Create a second asset
Reference the string
Open both assets
Close both assets
Reload the string table
Notice warnings in the log
Open the second asset
Notice missing strings