'Isothermal' means at constant temperature. In a strict sense, an isothermal process must be a reversible process because by definition, if every part of the system is at the same, constant temperature throughout the process, there can be no frictional or other irreversible effects giving rise to heat and causing local changes in temperature [see, for example, Rogers and Mayhew (1992)]. No real process can be perfectly isothermal, some come very close, especially if it is accepted that it is only the spatially-averaged temperature which must remain constant.
In processes operating on a single phase, heat transfer will result in a change in temperature unless exactly balanced by some other energy transfer, e.g., work, and this balance can be very difficult to achieve in practice. One solution might be to eliminate the heat transfer: but in reality, insulation can reduce heat transfer but cannot stop heat transfer completely. An alternative approach is to use systems which can accept some heat transfer without a change in temperature.
In two-phase systems, heat transfer can be accommodated without changing the temperature by altering the relative amounts of the two phases present. The most common example is a phase equilibrium in a pure substance at constant pressure. Ice in water is frequently used as a fixed point for temperature because this system remains at a constant temperature provided the pressure is constant and the rate of heat transfer is not sufficient to cause the system to depart significantly from equilibrium.
Rogers, G. F. C. and Mayhew, Y. R, (1992) Engineering Thermodynamics: Work and Heat Transfer: SI units 4th edn. Longman Group UK Ltd., Harlow, Essex. DOI: 10.1016/0009-2509(93)80061-T