Mix 1/2 cup of TSP into 2 gallons of warm water until completely dissolved. Use a sponge to clean the pickled oak wall unit with the TSP mixture, being careful not to saturate the wood. Rinse the unit with clean water and pat dry with a clean rag.
Dilute an oil-based paint with paint thinner, using 3 parts paint to 1 part thinner. Stir the thinner into the paint with a wooden stir stick until fully incorporated.
Apply paint to the oak wall unit, using a natural-bristle paintbrush. Cover a single section of the unit at a time, brushing on the paint in long, even strokes that follow the grain of the wood.
Wait 10 minutes for the paint to dry on the wood surface and penetrate into the pores.
Wipe the surface of the wood, using a piece of cloth to remove excess paint from the surface of the unit. Remove the paint in small increments until you have the level of transparency desired.
Even out the finish by brushing light or dark areas of the wood with a dry-bristle brush. Use the same long strokes used to apply the paint to even out the finish, spreading paint to blend the entire coating into a single hue. Clean off the bristles to keep them dry after every few strokes, using a dry piece of cloth. Allow the paint to dry overnight.
Cover the paint with two coats of flat, oil-based wood finish, using the paintbrush in the direction of the wood grain and slightly overlapping brush strokes for complete coverage. Wait about 15 minutes after the first finish layer before applying a second. Wait 48 hours for the finish to dry to the touch.