Fe(110) nanostripe arrays, consisting of alternating monolayer and double layer stripes, have been grown by step flow on vicinal W(110) substrates. The magnetic easy axis switches from in-plane in the monolayer to perpendicular in the double layer stripes. The data strongly suggest that magnetostatic interactions induce antiferromagnetic order in the double layer nanostripe array. It can be switched into a ferromagnetic arrangement by low external fields.