  Coherence is a defining feature of quantum condensates. These condensates are
inherently multimode phenomena and in the macroscopic limit it becomes
extremely difficult to resolve populations of individual modes and the
coherence between them. In this work we demonstrate non-equilibrium
Bose-Einstein condensation (BEC) of photons in a sculpted dye-filled
microcavity, where threshold is found for $8\pm 2$ photons. With this
nanocondensate we are able to measure occupancies and coherences of individual
energy levels of the bosonic field. Coherence of individual modes generally
increases with increasing photon number, but at the breakdown of thermal
equilibrium we observe multimode-condensation phase transitions wherein
coherence unexpectedly decreases with increasing population, suggesting that
the photons show strong inter-mode phase or number correlations despite the
absence of a direct nonlinearity. Experiments are well-matched to a detailed
non-equilibrium model. We find that microlaser and Bose-Einstein statistics
each describe complementary parts of our data and are limits of our model in
appropriate regimes, which informs the debate on the differences between the
two.
