To make muddy or to cover with mud.
From 'be-' plus 'muddy' (the adjective/verb), creating an intensified form meaning to make something very muddy or mud-covered.
Interestingly, 'bemuddy' is actually a variant of 'bemud'—English speakers created multiple versions of the same idea, and over centuries, standardization killed all but the most common ones (in this case, both died).
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