I have almost this exact situation. Frankly, my urine has never had this strange "pungent latex" stench until I became sexually active (6 or 7 months ago). It was almost immediate. But I practice safe sex, have only had one partner, and have returned to temporary celibacy for 2 months now, just to see if it makes any difference. The stench remains.
Out of morbid curiosity one time, I let my urine sit in the toilet one time, and when I returned half an hour later, the smell was far worse. It didnt even smell quite like urine anymore--- more like latex "gone bad". I dont even know how to describe it.
When I tried to bring it up with doctors, it got casually dismissed. Likely because I was being checked out for other issues (with my GP, I was visiting for a yeast infection at the time, so the doctor treated the urine situation as a by-product of that--which I dont buy at all. When I went to a separate walk-in women's health clinic months later, they ran tests for STIs, but they were all negative.)
Im going to try going back to the woman's clinic and demand a urine analysis that isnt assessed from the angle of screening for STIs, because it obviously cant be that. It's frustrating to worry over whether or not it's a sign of something far more serious and unrelated to sexual activity, and healthcare professionals approach it with such casualness.