The instanton approximation is a widely used approach to construct the semiclassical theory of tunneling. The instanton path bridges the regions that are not connected by classical dynamics, but the connection can be achieved only if the two regions have the same energy. This is a major obstacle when applying the instanton method to nonintegrable systems. Here we show that the ergodicity of complex orbits in the Julia set ensures the connection between arbitrary regions and thus provides an alternative to the instanton path in the nonintegrable system. This fact is verified using the ultra-near integrable system in which none of the visible structures inherent in nonintegrability exist in the classical phase space, yet nonmonotonic tunneling tails emerge in the corresponding wave functions. The simplicity of the complex phase space allows us to explore the origin of the nontrivial tunneling tails in terms of semiclassical analysis in the time domain. In particular, it is shown that not only the imaginary part but also the real part of the classical action plays a role in creating the characteristic step structure of the tunneling tail that appears as a result of the quantum resonance.