Place plastic sheeting on top of any landscaping that is next to or near the brick veneer to keep water and TSP solution off the foliage.
Spray the brick veneer thoroughly with water from a hose. Wear latex or work gloves and safety goggles to keep water and mud off your hands and out of your eyes. If the brick veneer is still muddy, proceed to the next step; otherwise, skip to step 6.
Fill a bucket with a 1/2 gallon of trisodium phosphate and a 1/2 gallon of water. Stir the mixture with a medium-stiffness nylon bristled brush. A medium-stiffness brush will not scratch the brick veneer. Trisodium phosphate is available at area hardware and home improvement stores and outlets.
Apply the TSP solution to any caked-on mud that spraying with water did not remove. Scrub the caked-on mud while you spray it with water.
Spray the brick veneer with water after scrubbing the surface to remove the TSP solution and any remaining mud. Continue spraying the brick veneer until it is mud free and clean.
Let the brick veneer air dry before removing the plastic sheeting covering the landscaping.