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I don't get the last sentence. Surely the only place it can flop (and thus be loose) is the hinge outside the radio casing? There should actually be a screw in this hinge - is this the one you've tried tightening?
No, it's not the hinge, that's nice and tight, it's actually the slot or hole where the antenna rod is inserted into the radio. I'm trying to find a picture of the back of a portable radio but I'm not having much luck.
This picture is the best I can find. See how there's a hinge in the antenna, yes, but there's also a short rod leading from the hinge into the radio itself? If that rod were loose and turning in its socket, instead of firm, the antenna would flop over instead of staying upright. My antenna is on top of the radio but essentially the same thing is happening.
My husband could answer this in a flash. Unfortunately, he's in Crete on a remote beach with no cell connectivity (unless he climbs up the hill), and, apparently, insufficient food and shelter.
Nevertheless, from what I've seen of this kind of thing, you can completely unscrew that part from where it's connected into the radio - unscrew the whole antenna and its "joint"... and likewise, screw it back in. So the obvious, most simplistic question, is is it just slightly unscrewed and therefore loose-ish? Don't hit me.
Next, if not that, are the threads stripped on the inside where it should screw in? If so, there's this super-thin teflon tape (?) that you can wrap around the screw-in part to create more bulk while still keeping the thread integrity. Or something. Anyway, I may be way off-base, but I've seen my husband deal with stuff that seems to me similar to what you are talking about.