Pretty straight forward.
I’ve often thought pool and bowling would be the most likely, but they obviously have separate leagues.
Are there any sports(that are not massively obscure) where men and women play competitively together?
Although I wouldn’t go as far to say that the NHL is an inter-gender league, Manon Rheaume played a few exhibition games for the Tampa Bay Lightning in the 90s.
Tennis and Curling both have mixed events.
I went to a Celtic event, and they were playing a game that looked like a cross between field hockey and lacrosse. I was surprised to see both men and women on the pitch. Anybody know what game I saw?
I’ve seen equestrian jumping events where male riders competed against female riders (and male horses against female horses!).
And, of course, female jockeys compete with male jockeys at many throroughbred race tracks.
I think Olympic shooting events are now segregated by gender, but that wasn’t always the case. In 1976, I remember that an American man and an American woman tied for the lead, and both wanted a playoff to determine the gold medalist. That didn’t happen, because the Olympic rules provided for a tiebreaking procedure without a playoff round, and the man was deemed the winner.
BUT… the neat part is, he considered her a co-champion, and he had her stand on the top step of the winners’ dais, and hold the gold medal with him while the national anthem played.
Hurling . Never seen women play it, though.
A bit obscure, but the best female fell-runners are in the same ballpark as the men and often compete in the one race. Fell running is the sport of running up and down hills or mountains. In the last everest marathon the top woman came 7th, for example.
OK, that was massively obscure 
Seeing as how she was almost 50 minutes after the winner, I think her high placing is an artifact of the low number of contestants rather than due to her ability being close to the best.
From following a link in this thread earlier today, I learned that Kyōtei (Japanese hydroplane racing) has both genders competing together, with parimutuel gambling allowed:
Poker.
Just a WAG, but chess?
Not that I’ve heard of any women grand-masters, but then I’ve only heard of Fisher and a few of the Russians.
You’ll be saying darts is a sport, next! 
I believe those crazy ultra-marathons are co-ed - and the women tend to win them, AIRC.
I used to think this as well, but somebody on here showed me otherwise. If I recall correctly the gap between men and women might be slightly smaller at very long distances, but men still win quite easily.
Which makes my WAG oh so true…
And if poker is a sport, chess sure is one aswell!
Hard to say whether it’s more theater or sport, but men and women do often compete on the same team in roller derby. These days, though, it seems that only all-female teams can draw a crowd.
Shouldn’t a “real sport” at least require a certain level of physical fitness and/or technique? If so, darts just sneaks in, but there’s no place for chess and poker! 
I´m not saying which is which and what is what. Just that if poker counts, then chess should as well.
But if you want to go down that slippery slope, defining sports… be my guest