Wavelength scale coherent optical sources are vital for a wide range of applications in nanophotonics ranging from metrology and sensing to nonlinear frequency generation and optical switching. In these respects, semiconductor nanowires (NWs) are of particular interest since they represent the ultimate limit of downscaling for photonic lasers with dielectric resonators. By virtue of their unique one-dimensional geometry NW-lasers combine ultra-high modal gain, support low-loss guided modes and facilitate low threshold lasing tuneable across the UV, visible and near infra-red spectral regions. Recently, optically pumped NW lasers have been demonstrated at room temperature and they can now be site-selectively integrated onto silicon substrates. While the fundamental carrier relaxation and gain dynamics of NW-lasers have been explored, the coherent dynamics have hitherto received comparatively little attention. In this contribution, we demonstrate that GaAs-AlGaAs core-shell nanowire lasers grown on silicon are capable of emitting pairs of phase-locked picosecond laser pulses when subject to incoherent pulsed optical excitation. By probing the two-pulse interference that emerges within the homogeneously broadened laser emission, we show that pulse pairs remain mutually coherent over timescales extending beyond ~30ps, much longer than the emitted laser pulse duration (~3 ps). Simulations performed by solving the optical Bloch equations produce good quantitative agreement with experiments, revealing how the phase information is stored in the gain medium close to transparency.