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We study the variational principle for discrete height functions (or equivalently domino tilings) where the underlying measure is perturbed by a random field. We show that the variational principle holds almost surely under the standard assumption that the random field is stationary and ergodic. The entropy functional in the variational principle homogenizes and is the same as for the uniform measure. Main ingredient in the argument is to show the existence, equivalence and characterization of the quenched and annealed surface tension. This is accomplished by a combination of the Kirszbraun theorem and two sub-additive ergodic theorems.