Power Supply Pumping
A highly simplified view of the problem is as follows. A class-D amplifier PWM modulates 2 output states. The first, sourcing power to the load. The second, commutating the load and reconstruction filter current. 

In a split supply half-bridge configuration the amplifier sources power from one supply rail and commutates through the other. At low loads or short-durations (high-frequency audio) the commutation current is easily absorbed into the bulk capacitance on the rail. 

In a full-bridge configuration the same supply rail that is sourcing power to the load is being commuated through. Hence the same supply that is sourcing load current also ends up being regenerated on the opposite PWM state. 

Additionally, some full-bridge class-d ampifiers will also commutuate the load through the top 2 or bottom 2 switching elements and not pump the supply at all during this 3rd PWM state.

If you were to operating a half-bridge amplifier without an LC reconstruction filter into a resistive load, there would be no "supply pumping". However the output waveform would not be continuous.