Reactive vs Real Power in DC-DC Converters
As the circuit charges the inductor, the current and voltage will be
  out of phase.


No, if you apply a voltage across an inductor you will get a current ramp. You are thinking in terms of AC sinusoidal situations and this is not one of them.


  you don't want real power being dissipated in the inductor and
  producing waste heat.


No, but you want the inductor to store energy in its magnetic field and this, can be seen as real power being forced into the inductor; it so happens that it doesn't turn to heat AND, importantly, can be liberated into the output circuit in the 2nd half of a typical switching cycle of a switching power converter.


  How can the two ways of analyzing the converter's power be reconciled?


Go back to basics: \$V = L\dfrac{di}{dt}\$

This can be used to: -


Derive the inductor current and energy used in a switching converter or, in applicably in switching supplies, it can be used to: -
Show that the current through an inductor is 90 degrees lagging the voltage across it when talking about sine wave excitation (inapplicable to switching converters but important in AC analysis of motors, transformers etc..)


It is the former that is helpful in the analysis of switching converters and not the latter.