The magnetic structures and the magnetic phase transitions in the Mn-doped orthoferrite TbFeO$_3$ studied using neutron powder diffraction are reported. Magnetic phase transitions are identified at $T^mathrm{Fe/Mn}_N approx$ 295~K where a paramagnetic-to-antiferromagnetic transition occurs in the Fe/Mn sublattice, $T^mathrm{Fe/Mn}_{SR} approx$ 26~K where a spin-reorientation transition occurs in the Fe/Mn sublattice and $T^mathrm{R}_N approx$ 2~K where Tb-ordering starts to manifest. At 295~K, the magnetic structure of the Fe/Mn sublattice in TbFe$_{0.5}$Mn$_{0.5}$O$_3$ belongs to the irreducible representation $Gamma_4$ ($G_xA_yF_z$ or $Pbnm$). A mixed-domain structure of ($Gamma_1 + Gamma_4$) is found at 250~K which remains stable down to the spin re-orientation transition at $T^mathrm{Fe/Mn}_{SR}approx$ 26~K. Below 26~K and above 250~K, the majority phase ($> 80%$) is that of $Gamma_4$. Below 10~K the high-temperature phase $Gamma_4$ remains stable till 2~K. At 2~K, Tb develops a magnetic moment value of 0.6(2)~$mu_mathrm{B}/$f.u. and orders long-range in $F_z$ compatible with the $Gamma_4$ representation. Our study confirms the magnetic phase transitions reported already in a single crystal of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ and, in addition, reveals the presence of mixed magnetic domains. The ratio of these magnetic domains as a function of temperature is estimated from Rietveld refinement of neutron diffraction data. Indications of short-range magnetic correlations are present in the low-$Q$ region of the neutron diffraction patterns at $T < T^mathrm{Fe/Mn}_{SR}$. These results should motivate further experimental work devoted to measure electric polarization and magnetocapacitance of TbFe$_{0.5}$Mn$_{0.5}$O$_3$.
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