This question has been studied in some detail for instantons in QM, for example the standard instanton in the potential $V=(x^2-x_0^2)^2$. Naive analytic continuation $\tau\to it$ gives a complex trajectory which does not have direct physical meaning, but corresponds to a saddle of the path integral in the same way that ordinary real integrals have complex saddle points.
This does not mean that the instanton does not have physical significance. The instanton calculation provides a semi-classical approximation to the path integral, and there is a one-to-one map to the WKB approximation for the Schroedinger equation. In particular, we can write down the WKB wave function for a particle tunneling from one minimum to the other, and this solution corresponds to instanton contribution in euclidean path integral.