Vegetable broth, onions, celery, carrots, and garlic
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The Food Lab's Kenji tested soaking a turkey in broth: http://www.seriouseats...
Basically, he found that it didn't make a difference in flavor, and that soaking in a solution without salt resulted in meat that was drier than meat that hadn't been soaked in anything. So if you can't brine, then I'd suggest not soaking at all. Have you thought about dry brining?
Lisanne is a trusted home cook.
Depending on the reason for the veto (is someone cutting out salt for medical reasons, for example), if salt is not a problem, I find a dry brine a few days ahead of time makes for a delicious turkey: https://food52.com/recipes...
BerryBaby is trusted source on General Cooking
I have never brined our turkey wet or dry. Always juicy and moist.