The orientation of the wires in a speaker cable is one of the biggest factors that determines the "sound" of that cable, so it is logical to conclude that braiding the cables will affect the sound.
The alternating currents that flow through speaker wires create moving magnetic fields. These fields cut through nearby conductors and induce currents in them, the transformer effect. The amount of current induced is proportional to the amount of current inducing it and the proximity and orientation of the wires. So it doesn't really matter if they carry the same signal, the wires will interact and this interaction will be affected by braiding them. In addition, the capacitance and inductance of the cables will be changed.
Also, tri-wire cables do not all carry the same signal. Some drivers draw more current than others so the magnetic fields developed around these cables would be greater than those that carry less current.
Whether all of this will be audible in your system is impossible to predict.
The alternating currents that flow through speaker wires create moving magnetic fields. These fields cut through nearby conductors and induce currents in them, the transformer effect. The amount of current induced is proportional to the amount of current inducing it and the proximity and orientation of the wires. So it doesn't really matter if they carry the same signal, the wires will interact and this interaction will be affected by braiding them. In addition, the capacitance and inductance of the cables will be changed.
Also, tri-wire cables do not all carry the same signal. Some drivers draw more current than others so the magnetic fields developed around these cables would be greater than those that carry less current.
Whether all of this will be audible in your system is impossible to predict.