  The miscibility of two interacting quantum systems is an important testing
ground for the understanding of complex quantum systems. Two-component
Bose-Einstein condensates enable the investigation of this scenario in a
particularly well controlled setting. In a homogeneous system, the transition
between mixed and separated phases is fully characterised by a `miscibility
parameter', based on the ratio of intra- to inter-species interaction
strengths. Here we show, however, that this parameter is no longer the optimal
one for trapped gases, for which the location of the phase boundary depends
critically on atom numbers. We demonstrate how monitoring of damping rates and
frequencies of dipole oscillations enables the experimental mapping of the
phase diagram by numerical implementation of a fully self-consistent
finite-temperature kinetic theory for binary condensates. The change in damping
rate is explained in terms of surface oscillation in the immiscible regime, and
counterflow instability in the miscible regime, with collisions becoming only
important in the long time evolution.
