  There has been a growing interest in non-Hermitian physics. One of its main
goals is to engineer dissipation and to explore ensuing functionality. In
magnonics, the effect of dissipation due to local damping on magnon transport
has been explored. However, the effects of non-local damping on the magnonic
analog of the Josephson effect remain missing, despite that non-local damping
is inevitable and has been playing a central role in magnonics. Here, we
uncover theoretically that a surprisingly rich dynamics can emerge in magnetic
junctions due to intrinsic non-local damping, using analytical and numerical
methods. In particular, under microwave pumping, we show that coherent spin
precession in the right and left insulating ferromagnet (FM) of the junction
becomes synchronized by non-local damping and thereby a magnonic analog of the
$\varphi$ Josephson junction emerges, where $\varphi$ stands here for the
relative precession phase of right and left FM in the stationary limit.
Remarkably, $\varphi$ decreases monotonically from $ \pi$ to $\pi/2$ as the
magnon-magnon interaction, arising from spin anisotropies, increases. Moreover,
we also find a magnonic diode effect giving rise to rectification of magnon
currents. Our predictions are readily testable with current device and
measurement technologies at room temperatures.
