  Accreting millisecond X-ray pulsars (AMXPs) show burst oscillations during
thermonuclear explosions of the accreted plasma which are markedly different
from those observed in non-pulsating low mass X-ray binaries. The AMXP XTE
J1814-338 is known for having burst oscillations that are phase locked
(constant phase difference) and coincident with the accretion powered
pulsations during all its thermonuclear bursts but the last one. In this work
we use a coherent timing analysis to investigate this phenomenon in more detail
and with higher time resolution than was done in the past. We confirm that the
burst oscillation phases are, on average, phase locked to the accretion powered
pulsations. However, they also display moderate (<~ 0.1 cycles) drifts during
each individual burst, showing a repeating pattern that is consistently
observed according to the thermonuclear burst phase (rise, peak, tail). Despite
the existence of these drifting patterns, the burst oscillation phases somehow
are able to average out at almost the exact position of the accretion powered
pulsations. We provide a kinematic description of the phenomenon and review the
existing models in the literature. The phenomenon remains without a clear
explanation, but we can place important constraints on the thermonuclear burst
mechanism. In particular, the observations imply that the ignition point of the
thermonuclear burst occurs close to the foot of the accretion column. We
speculate that the burning fluid expands in a backward tilted accretion column
trapped by the magnetic field, while at the same time the burning flame covers
the surface.
