How can a non-EU citizen leave Belgium while awaiting extension of their Belgian residence permit?

This question is a follow-up to a query by @Traveler:

I promised my girlfriend to take her to see the Northern Lights. We live in Belgium and because of the visa application she cannot leave the country for 6 months…

How can such a Belgian resident travel to other countries – both Schengen and outside the Schengen area? Is there a temporary visa that Belgian authorities could issue in this scenario? Or perhaps is it possible to apply for a separate Schengen visa from another Schengen country?