Why is there a systematic bias in the Asian Monsoon in the Met Office Unified Model?

Many global-climate models have substantial biases in their predictions of the Asian monsoon. For example, the Met Office Unified Model predicts a  monsoon trough that is too zonal and therefore underestimates summer rainfall over south and east Asia. These errors have persisted over many cycles of research-to-operations, and appear robust to significant developments in all major parametrizations in the model. Here, we address a simple question: why are these biases systematic? That is, why have they not been removed by optimization of parameters in the model's physics? Using a Perturbed Parameter Ensemble of AMIP simulations, we show that a strong constraint exists which prevents the Unified Model from simultaneously producing an unbiased monsoon and unbiased global top-of-atmosphere radiation fluxes. We use this constraint to define a scalar parameter, the "structural bias"  of the ensemble, the magnitude of which measures the conflict between the constraints and therefore how "untunable" the model is. We identify the drivers of this parameter, show that it is related to an inability to independently affect the properties of tropical and extra-tropical clouds, and suggest ways in which it could be reduced in future model versions.