We investigate the role of asymmetry in the stationary measures of integrable probabilistic models with boundary interactions. This paper is the first in a series of three focusing on the open asymmetric simple exclusion process (ASEP). The stationary measure of open ASEP admits a two-layer Gibbs representation in which a correction term, which we call the Bryc polynomial, encodes the dependence on the asymmetry parameter. We give a probabilistic interpretation of the Bryc polynomial in terms of a birth-death-type Markov chain. As an application, we obtain sharp total variation approximations of the stationary measure by Bernoulli product measures in the high- and low-density phases of the shock region. These approximations yield further applications, including moderate deviation estimates for the stationary measure of open ASEP, as well as asymptotic results for the related model of open ASEP with light particles.
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