There are napkins which you place over your clothing, covering you from your neck down, if draped well covering all under the way of your food, from your plate to your mouth.
One such I have seen was a big square of cotton or linen, made into a traditional napkin, with a button hole made in one corner. The napkin would be spread on the front of the person and a shirt button would come through the button hole just below the neck. The fabric being quite sturdy would stay where it was spread. Alternate attachment methods are clasps on a chain going round the neck, or clipping the napkins to the front of the clothing.
Starch in your fabric napkin will help in it staying in place, as it will make the fabric more stiff.
Not having clothing in the way.
If you can chose where and when you eat, very juicy fruits are best eaten ready to get into the shower, while the water is already running to get to temperature.
Alternatively, eat when you are already in the bath, although a bath with a shower is advised as you may want to rinse off when coming out of the water.
I personally do not feel it as good a solution for hot food, like meats which drip sauce, certainly not while they are still hot.
But a shower just after eating will clean your body all over.
This does not keep you clean, it will clean you up after getting dirty.
As a last option:
Refrain from eating foods that would get you into trouble. This is what notable people do in public. Some going to the extend not eating anything in public.