Mixed-phase filters blend the strengths of linear-phase FIR filters (flat phase response, precise magnitude control) and minimum-phase IIR filters (low latency, no pre-ringing). The low frequencies are corrected with minimum-phase filters (where pre-ringing would be most audible), while high frequencies use linear-phase filters (where timing precision matters more).
This is the approach used by advanced room correction systems like Dirac Live and Trinnov. Pure linear-phase correction sounds most accurate but adds too much latency for real-time use. Mixed-phase provides 90% of the benefit with acceptable latency.