Geometric last passage percolation on a strip of width N has a unique stationary measure, given by Barraquand, Corwin and Yang as a marginal of an explicit two-layer weight. I will describe two scaling limits of it. Thinning the bulk gives a pair of Poisson processes conditioned on a ballot event, with discrete Bessel marginals. Widening the strip gives a stationary measure of the half-space model, with family parameter $s = \max\{1, c_1, c_2\}$ for all boundary parameters. The two limits turn out to be the same object seen at two scales, by Pitman's 2M-X theorem.