We present detailed investigations in single crystals of two recently reported quaternary intermetallic compounds EuRhAl$_4$Si$_2$ and EuIrAl$_4$Si$_2$ employing magnetization, electrical resistivity in zero and applied fields, heat capacity and $^{151}$Eu M{o}ssbauer spectroscopy measurements. The two compounds order antiferromagnetically at $T_{rm N1}$ = 11.7 and 14.7,K, respectively, each undergoing two magnetic transitions: the first from paramagnetic to incommensurate modulated antiferromagnetic, the second at lower temperature to a commensurate antiferromagnetic phase as confirmed by heat capacity and M{o}ssbauer spectra. The magnetic properties in the ordered state present a large anisotropy despite Eu$^{2+}$ being an $S$-state ion for which the single-ion anisotropy is expected to be weak. Two features in the magnetization measured along the $c$-axis are prominent. At 1.8,K, a ferromagnetic-like jump occurs at very low field to a value one third of the saturation magnetization (1/3 M$_0$) followed by a wide plateau up to 2,T for T = Rh and 4,T for T = Ir. At this field value, a sharp hysteretic spin-flop transition occurs to a fully saturated state (M$_0$). Surprisingly, the magnetization does not return to origin when the field is reduced to zero in the return cycle, as expected in an antiferromagnet. Instead, a remnant magnetization 1/3 M$_0$ is observed and the magnetic loop around the origin shows hysteresis. This suggests that the zero field magnetic structure has a ferromagnetic component, and we present a model with up to third neighbor exchange and dipolar interaction which reproduces the magnetization curves and hints to an up-up-down magnetic structure in zero field.
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