Am Std Jaguar Bass w/ split coil - How does the tone compare to an actual P Bass and the MIJ Jags?

I've got a MIJ Jaguar right now, and I think it's sounds pretty sweet through my rig - though a little 'flubby' and hollow/woody sounding though when I play with a pick, and put it through my Radial DI for recording.

I've been toying with getting a precision bass, but I do love the vibe and feel of the Jaguar, and am tempted by the new American Standard Jaguar basses with the Precision pickup. I'd love to hear your guys' thoughts on how it the Jags compare tone wise to a real P Bass.

Thanks!