X-ray timing of the accretion-powered pulsations during the 2003 outburst of the accreting millisecond pulsar XTE J1814-338 reveals variation in the pulse time of arrival residuals. These can be interpreted in several ways, including spin-down and wandering of the fuel impact point around the magnetic pole. In this Letter we show that the burst oscillations of this source are coherent with the persistent pulsations, to the level where they track all of the observed variation in the residuals. Only one burst, which occurs at the lowest accretion rates, shows a significant phase offset. We discuss what might lead to such rigid phase-locking between the modulations in the accretion and thermonuclear burst emission, and consider the implications for spin variation and the burst oscillation mechanism. Premature ignition due to higher local temperatures at the fuel stream impact point may play a role in generating the highly unusual burst oscillations of this source.