The Taylor-phase complex intermetallic compound T-Al 3 Mn, its solid solutions with Pd and Fe, T-Al 3 ͑Mn, Pd͒ and T-Al 3 ͑Mn, Fe͒, and the related decagonal d-Al-Mn-Fe quasicrystal belong to the class of magnetically frustrated spin systems that exhibit rich out-of-equilibrium spin dynamics in the nonergodic phase below the spin-freezing temperature T f . Performing large variety of magnetic experiments as a function of temperature, magnetic field, aging time t w , and different thermal histories, we investigated broken-ergodicity phenomena of ͑i͒ a difference in the field-cooled and zero-field-cooled susceptibilities, ͑ii͒ the frequencydependent freezing temperature, T f ͑͒, ͑iii͒ hysteresis and remanence, ͑iv͒ ultraslow decay of the thermoremanent magnetization, ͑v͒ the memory effect ͑a state of the spin system reached upon isothermal aging can be retrieved after a negative temperature cycle͒, and ͑vi͒ "rejuvenation" ͑small positive temperature cycle within the nonergodic phase erases the effect of previous aging͒. We show that the phenomena involving isothermal aging periods ͑the memory effect, rejuvenation, and the ultraslow decay of the thermoremanent magnetization͒ get simple explanation by considering that during aging under steady external conditions, localized spin regions quasiequilibrate into more stable configurations, so that higher thermal energy is needed to destroy these regions by spin flipping, as compared to the thermal energy required to reverse a frustrated spin in a disordered spin-glass configuration that forms in the case of no aging. Common to all the investigated brokenergodicity phenomena is the slow approach of a magnetically frustrated spin system toward a global equilibrium, which can never be reached on accessible experimental time scales due to macroscopic equilibration times.