Re-mixing high and low pass filtered audio has interference in the shared transition band?
You are assuming that a parallel high pass and low pass add up to "flat". In most cases, they do not. You have to carefully design both the phase and the amplitude responses in the transition band to match each other.

There are a few classes of filters that are useful here


1st order lowpass/highpass to add up to unity
Odd order butterworth filters add up to an allpass filter (flat frequency response, but some phase distortion). The two paths are either 90 or 270 degrees phase shifted (depending on order)
Linkwitz Riley filter (two butterworth cascaded) also add up to an allpass (when choosing the appropriate sign for recombination). The two paths are in phase or out of phase (depending on order).
linear phase FIR filters that are designed complementary. 
Specific types of filter banks (reconstructing)


All of this only helps if your processing doesn't induce significant differential phase shifts. Unless this phase shift is constant and can be compensated, you will always get some amount of interference.