  Spatial quantum coherence between two separated driven-dissipative polariton
condensates created non-resonantly and with a different occupation is studied.
We identify the regions where the condensates remain coherent with the phase
difference continuously changing with the pumping imbalance and the regions
where each condensate acquires its own chemical potential with phase
differences exhibiting time-dependent oscillations. We show that in the mutual
coherence limit the coupling consists of two competing contributions: a
symmetric Heisenberg exchange and the Dzyloshinskii-Moriya asymmetric
interactions that enable a continuous tuning of the phase relation across the
dyad and derive analytic expressions for these types of interactions. The
introduction of non-equal pumping increases the complexity of the type of the
problems that can be solved by polariton condensates arranged in a graph
configuration. If equally pumped polaritons condensates arrange their phases to
solve the constrained quadratic minimisation problem with a real symmetric
matrix, the non-equally pumped condensates solve that problem for a general
Hermitian matrix.
