Class B amplifiers tend to be efficient but suffer from high distortion when used with small signals.
It was therefore natural to try to develop compromise designs. So-called "Class AB" has the same topology as a class B amplifier, but the bias is modified such that both devices are active concurrently over part of the amplifiers range.
Such a circuit behaves as a class A amplifier in the region where both both devices are in the linear region, however the circuit cannot strictly be called class A if the signal passes outside this region, since beyond that point only one device will remain in its linear region and the transients typical of class B operation will occur.
However, for many applications (such as amplifying music), Class AB is widely considered a good compromise, since much of the time the music is quiet enough that the signal stays in the "class A" region, where it is reproduced with good fidelity, and by definition if passing out of this region, is large enough that the distortion products typical of class B are relatively small.