Enthalpy as a path-independent state function

Today in class we defined the enthalpy:

H = U + P V

We found that for an infinitely slow heating process at constant pressure, all of the heat that was added was accounted for a change in enthalpy:

Delta H^t = Q

I was asked how the enthalpy could be path-independent, when it is equal to Q, which is path-dependent. The reason is that the above equation only applies to the specific process of infinitely slow heating. If we do not heat in such a way that we are at equilibrium during the entire process, then Q is not necessarily accounted for by the change in enthalpy.

For a fast heating rate, that results in the same final state, the change in enthalpy will be the same (since it is a state function), but the value of Q can be different from it.