Supply priests, interim pastors, etc. My parish has not had a rector for about 12 months. There are a pool of priests in the Diocese that you can pay to come in on Sundays to run the services. Some are retired, some have secular careers, some have non-parish based ministries (such as the priest that runs the Cathedral Shelter in Chicago).
Of course, that means that there is no full time or part time pastor present to handle the pastoral needs of the parishioners. Lay ministers handle some of this, and some of the supply priests that are local handle some of this. It's not the best solution, but you can keep going if your parish is determined to do so.
I'm familar with the methodology. We endured a six months interim run of assorted types while going through our last search some years back..But my question was morephilosophical..the minister is the soul, the backbone of the parish..OTOH..perhaps it speaks more to the spirituality of the laity that kept it going..