Magnetization and specific-heat measurements on a UIrSi₃ single crystal reveal Ising-like antiferromagnetism below T_N = 41.7K with the easy magnetization direction along the c axis of the tetragonal structure. The antiferromagnetic ordering is suppressed by magnetic fields > H_c (μ₀H_c = 7.3 T at 2 K) applied along the c axis. The first-order metamagnetic transition at H_c exhibits asymmetric hysteresis reflecting a slow reentry of the complex ground-state antiferromagnetic structure with decreasing field. The hysteresis narrows with increasing temperature and vanishes at 28 K. A second-order metamagnetic transition is observed at higher temperatures. The point of change of the order of transition in the established H-T magnetic phase diagram is considered as the tricritical point (at T_tc = 28 K and μ₀H_tc = 5.8 T). The modified-Curie-Weiss law fits of temperature dependence of the a- and c-axis susceptibility provide opposite signs of Weiss temperatures, Θ^a_p  ∼  -51K and Θ^c_p ∼  +38K, respectively. This result and the small value of μ₀H_c contrasting to the high TN indicate competing ferromagnetic and antiferromagnetic interactions responsible for the complex antiferromagnetic ground state. The simultaneous electronic-structure calculations focused on the total energy of ferromagnetic and various antiferromagnetic states, the U magnetic moment, and magnetocrystalline anisotropy provide results consistent with experimental findings and the suggested physical picture of the system.