I'm going to be contrarian, so please don't be upset, but:
I've owned and heavily gigged/recorded a few different P/J basses over the years (Spector, Fender, Ibanez). I was always looking for a "best of both worlds" tone. In my experience, they can do P-Bass really well, and they can do "bridge-soloed J Bass" really well, but the blended tone was always just a little weird. Maybe out of phase, maybe unequal output, maybe something entirely different -- I don't know. But I was always disappointed (regardless of brand) with that "ultimate tone" pursuit, and I've found that when I'm blending pickups, I VASTLY prefer a matched set, whether it be soapbars, J-pickups, or MM-style humbuckers.
So IME/IMO, a P/J is a great option for versatility if you're a guy who wants to used a P-Bass most of the time but has the need to switch to Jaco-jazzy for a few songs (or gigs), but I'm willing to be that if you're totally honest with your ears, you'll find a bigger and better overall tone by simply committing to a particular style of pickup config instead of splitting the difference.