Is it possible to design linear-phase filters that sum to a flat frequency response?
Yes, of course.
If so is it practical to use them in real-time audio processing for as many as 10 bands?
That depends on your definition of practical. The problem at audio is that FIR filters at low frequencies get really long. It depends on the lowest frequency, desired steepness and amount cross band rejection. but a few thousand taps is typical. These can be fairly efficiently implemented using an FFT based scheme like Overlap Add, but that introduces a lot of latency: typically it's twice the filter length. That's prohibitive for many real time applications.
If you want to build a standard 10-band audio equalizer: this can be done very efficiently with cascaded biquads where each biquad is a "peakingEQ" filter and simply recalculate the filter coefficients whenever a band changes gain.
Parallel IIR bandpass filters are difficult to phase manage but it's possible. Odd Butterworth and Linkwitz Riley filter sum to a flat magnitude response but NOT a flat phase response, so you need to put compensation allpass filters in the parallel paths