Shirt stays? You mean something that keeps your shirt pulled down it looks nice and tight, so the lines in the back look sharp?
Way back in the Regiment, two methods were generally used. The cheap way was to use boot blousers (name?), link a few of them together, and use them to maintain tension .....and I'm sure leave marks on the body if improperally done. The expensive way, initially anyhow, was to buy what they called a baseball belt.
Frankly, once out of the Regiment and in the regular Navy, I didn't bother ..... but on the other hand, the senior chief did take me aside one day and said that the troops expect their officers to look sharp (I wasn't even getting my shirts with military creases). So take it with a grain of salt.
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("I'm gonna teach you the meaning of pain."--Jacoby
"You like pain?" [hits pirate in the head with a pole] "Try wearing a corset."--Elizabeth, (w,stte), "POTC")