a known effect, which was previously studied in other forms [@book; @NaturePhot2]. A new phenomenon reported here is the SSB of leaky modes, which include nonvanishing tails extending into the free space outside of the DWP structure. Even if the tails have small amplitudes, it is interesting to analyze their structure in asymmetric modes, as this issue was not considered previously. To this end, we here focus on the symmetric and asymmetric states in the setting based on the DWP (\[potential\]) with $A=0.5$ and $\Lambda =14$, embedded into a broad domain of size $L=128$, see Eqs. (\[potential\]), (\[P2\]), and (\[bc\]). In this case, the asymmetric modes are found at $k\geq -0.100$.

A characteristic example of *asymmetric tails*, i.e., left and right ones with unequal amplitudes, is shown in Fig. \[tails\](a) for $k=-0.075$. Further, separately calculated total norms of the right and left tails, in regions $\Lambda /2<x<L/2$ and $-\Lambda /2<x<-L/2$, along with the norm of the tails in the coexisting unstable symmetric leaky mode, are displayed, as functions of $k$, in Fig. \[tails\](b). This dependence exhibits two notable features. First, the asymmetry between the right and left tails emerges at $k=-0.100$ and gradually increases with the increase of $k$ (i.e., 