Just putting my thought out: aren’t passive capacitors and coils (C an L), and maybe even resistors causing a PHASE shift on passive crossovers, and NOT so using active crossovers?
This happens with active and passive crossovers. it is the natural outcome of using analog circuits to create high and low pass circuits as well as any equalizers. It is not distortion. The concern in either case is that the combined (electrical + acoustical) amplitude and phase match seamlessly across the crossover band.
In terms of design, crossover designers are creating to systems, drivers and circuits, which sum to a final result. Changes in amplitude go hand in hand with phase shifts. The one area where things get easier is in the active crossover we can ignore driver impedance as part of the equation. We pay attention strictly to driver amplitude and phase.