In this paper, we study the asymptotic variance of sample path averages for inhomogeneous Markov chains that evolve alternatingly according to two different π-reversible Markov transition kernels P and Q. More specifically, our main result allows us to compare directly the asymptotic variances of two inhomogeneous Markov chains associated with different kernels Pi and Qi, i ∈ {0, 1}, as soon as the kernels of each pair (P0, P1) and (Q0, Q1) can be ordered in the sense of lag-one autocovariance. As an important application, we use this result for comparing different data-augmentation-type Metropolis- Hastings algorithms. In particular, we compare some pseudo-marginal algorithms and propose a novel exact algorithm, referred to as the random refreshment algorithm, which is more efficient, in terms of asymptotic variance, than the Grouped Independence Metropolis- Hastings algorithm and has a computational complexity that does not exceed that of the Monte Carlo Within Metropolis algorithm