Just a quick primer on what may be happening. The outside coil has air pushed through it so that it exits the outside back of the unit. This air is pulled in through the ventilation ribs built into the outside of the cabinet. Along with the air is the "stuff" floating in the air which can include pollen, dirt (of many different types), floating seeds from trees and plants, concrete dust if there was construction going on and the list goes on. Many window units use the outside fan to pick up condensate water and fling/blow that through the coil to gain a bit of added efficiency. You take the mix of air-born things and add a little moisture and heat (that coil gets hot) and you've got a perfect setup for making some really unique "cement" or "mud" as you called it .
You want to clean these by "flushing" from the outside to the inside of the coil. Trouble is when you do this, and the crud starts coming off, there is really no good way for that to be drained out unless the unit you have has a plug that allows that section of the bottom of the case to be drained easily. If you still have the old units experiment with different household cleaning fluids to find which one best dissolves the "cement" bond. A hand spray bottle should be all you need and of course patience. Each area of the country and even your city may have different components that make up the problem. Good Luck!