  Motile cilia beat in an asymmetric fashion in order to propel the surrounding
fluid. When many cilia are located on a surface, their beating can synchronise
such that their phases form metachronal waves. Here, we computationally study a
model where each cilium is represented as a spherical particle, moving along a
tilted trajectory with a position-dependent active driving force and a
position-dependent internal drag coefficient. The model thus takes into account
all the essential broken symmetries of the ciliary beat. We show that taking
into account the near-field hydrodynamic interactions, the effective coupling
between cilia can become nonreciprocal: the phase of a cilium is more strongly
affected by an adjacent cilium on one side than by a cilium at the same
distance in the opposite direction. As a result, synchronisation starts from a
seed at the edge of a group of cilia and propagates rapidly across the system,
leading to a synchronisation time that scales proportionally to the linear
dimension of the system. We show that a ciliary carpet is characterised by
three different velocities: the velocity of fluid transport, the phase velocity
of metachronal waves and the group velocity of order propagation. Unlike in
systems with reciprocal coupling, boundary effects are not detrimental for
synchronisation, but rather enable the formation of the initial seed.
