When we think of "mud-caked," words like dirty, grimy, and muddy come to mind. However, there are a variety of other synonyms that can be used to describe the situation. For example, words like soiled, stained, and sullied could all be used interchangeably. Other options include smudged, besmirched, and besmeared. While these may not be as commonly used as dirty or muddy, they can add variety to your writing and help create a more vivid image in the reader's mind. Ultimately, the choice of synonym will depend on the context and tone of the piece, but there are certainly many options to choose from.