+1 to the synthetic workout shirts being problematic. What my internet sleuthing turned up, and this bore out in practice, is that it is the synthetic + an antiperspirant (vs deodorant) that causes the problem. It makes the fabric hold the smell a lot more, and won't give it up. If you have a synthetic fabric that gets that weird almost sweet/fruity BO odor, before you're really sweat enough to be getting a big BO smell? That's the reaction from that.
Solution: I replaced all my synthetic shirts (RIP), and bought new ones, which I never use with antiperspirant. Haven't had the issue since. I mean, they still get smelly, obviously, but just normal person smelly which then washes out.
I've also heard the yellow staining on shirts only happens with sweat + antiperspirant as well, that it's not actually the sweat staining it. I have not experimented with this, though, as neither DH or I have had the yellow pit dilemma.
Final note: be aware that medications and dietary changes can affect body odor. I was recently worried my synthetic shirt stank was coming back, but on ALL my clothes. Turns out, a new medication I was on made me smell like a teenage boy's gym bag. No bueno.