Physics conundrum

[bbi5291]

This is quite a well-known little puzzler, and some of you undoubtedly will have seen it before.
Suppose that the same current is flowing in the same direction in two long, straight, parallel wires, and that this current exists in the form of electrons moving with constant velocity, which is the same in both wires. (We observe from an inertial frame.) We know how to calculate the attractive force between the two lines with the Biot-Savart law. Now then, we change our frame of reference to that (also inertial) of one of the moving electrons. In this frame of reference, why is there still an attractive force between the two wires?
You may be tempted to say that there are also electrons stationary with respect to the wire, which now appear to be moving backward and hence are responsible for an attractive magnetic force. However, that is not a satisfactory explanation, since it changes the bodies on which forces act.