What does it mean if a economic model is internally consistent? What happens if it is not? Is it not closed anymore? Can it still be solved? Does it still have an equilibrium?
In theoretical modeling the consistency is applied in the same way as in philosophy/logic. Internal consistency simply means that the argument is consistent with itself and has no contradiction within itself (as opposed to external consistency where its not enough for argument to be valid on its own but it should also not contradict other facts).
A simple verbal example of internal inconsistency would be statement: "I always lie". The statement is internally inconsistent because if I always lie then the statement above must be also lie. But if the statement is lie that implies I do not always lie so it is not correct and it contradicts itself.
Math can reveal if a model is internally inconsistent. If you have model relaying on following system: $$a+b =1$$ $$5a+5b=15$$
You would get that $5 = 0$ which means that the model is clearly inconsistent with itself. However, sometimes the model might be internally inconsistent because the assumptions made in the model are not consistent with each other but the model might still solve and even have equilibrium (if the logical fallacy is made in a step where you build equations based on the assumptions - or make assumptions that are not consistent with each other but the consistency of assumptions is not checked) and finding the inconsistency is harder.
Moreover, note that internal consistency of model has also some special meaning in statistics, but from your question I assume that was not the meaning you were looking for.