We study the sensitivity of coupled condensate formation dynamics on the history of initial stochastic domain formation in the context of instantaneously quenched elongated harmonically-trapped immiscible two-component atomic Bose gases. The spontaneous generation of defects in the fastest condensing component, and subsequent coarse-graining dynamics, can lead to a deep oscillating microtrap into which the other component condenses, thereby establishing a long-lived composite defect in the form of a dark-bright solitary wave. We numerically map out diverse key aspects of these competing growth dynamics, focussing on the role of shot-to-shot fluctuations and global parameter changes (initial state choices, quench parameters and condensate growth rates). We conclude that phase-separated structures observable on experimental timescales are likely to be metastable states whose form is influenced by the stability and dynamics of the spontaneously-emerging dark-bright solitary wave.
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