Use of the LMTD concept requires that the heat capacity (m.c_p) can be regarded as constant, as well as the overall heat transfer coefficient (U). For phase-change in parts of the heat exchanger both of these are violated, hence LMTD has no meaning.
For simple calculations, one might (as Leander suggests), ignore the superheating/subooling regions if the major part of the heat duty is in the phase-change region. However, this might sometimes lead to serious errors, especially when you are cooling a superheated vapor (small amount of heat, but also a low heat transfer coefficient) . A different approach would be to split the heat exchanger into separate regions, each having approximatel constant (m.c_p) and U, and then calculate area requirement for each zone. The sum of areas is then an estimate of the total heat exchanger surface requirement.