Using first-principles calculations, we investigate electronic structures of α arsenic phosphide under strain. It is a two-dimensional monolayer composed of an equimolar mixture of phosphorus and arsenic, whose multilayer correspondents were synthesized very recently. According to structure optimizations and phonon calculations, the α phase branches into three distinct allotropes. Monolayers of the α1 and α3 phases are direct-gap semiconductors with band gaps that are similar to that of the α phosphorene. They exhibit anisotrpic carrier mobility. Specifically, the α3 phase exhibits the electron mobility of ∼10 000 cm2 V–1 s–1, which is 1 order of magnitude larger than that for the α phosphorene. Likewise, their electronic structures display highly anisotropic behavior under strain different from that of the α phosphorene. The complex response under strain can be mostly understood in terms of the relative alignment of bonding and antibonding As–P states under a specific strain.