Measure the Phase Difference
The problem doesn't mention any LP filter or any way the average voltage is actually generated (as you noted).
It just asks what will be the average voltage (i.e. you don't have to assume any LP circuit; average voltage of a signal is defined even if there is no circuit actually generating the average voltage).
In other words: What will be the duty cycle of the output of the XOR gate? Multiply that by \$V\$ and you get the average voltage.