I just posted this same information in a previous thread....
From my experience I've found that the reason the brine turns milky is because it is either melting or leeching your cheese. The cause is that the acidity of the brine is completely different from the acidity of your cheese.
The solution to this problem is really simple...
Simply save the whey from the cheese you make and produce your brine solution from this whey. That way your cheese and the brine solution will be of the same acidity.
I use this method religiously during the manufacture of Gouda, Mozzerella, Provolone and Swiss (except when I forget). Since I've been doing this, I have not had any problem with my cheese melting or leeching out into the brine solution.
Hope this helps.