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We report a distinct interlayer antiferromagnetic stacking order in Sr₂IrO₄ thin films when substantially large orthorhombic distortion (> 1.5%) is applied via biaxial lattice strain in Sr₂IrO₄/Ca₃Ru₂O₇ heterostructures. The anisotropic biaxial strain lowers the crystal symmetry of Sr₂IrO₄ from tetragonal to orthorhombic, stabilizing the J_eff = ½ pseudospin moment direction via magnetic anisotropy while evoking the competition between interlayer exchange and pseudo-dipolar interactions. Using resonant X-ray magnetic scattering, we observed La₂CuO₄-like interlayer antiferromagnetic order where the magnetic moment is parallel to the elongated b-axis and its stacking pattern is inverted from that of the tetragonal Sr₂IrO₄ crystal. This interlayer stacking order is explained by the competition between interlayer interactions such that the nearest interlayer exchange interaction exceeds the interlayer pseudo-dipolar interaction. Our result suggests that anisotropic strain-induced orthorhombic distortion provides a delicate knob for tuning the long-range magnetic order and enhancing the magnetic quantum fluctuations in quasi-two-dimensional systems.