You need a pair of "splatchers" - these are (home-made) wooden boards that strap onto the feet over rubber boots to allow the wearer to walk over soft mud without the feet sinking in. I believe they are reasonably common amongst yachties in the tidal reaches of the Thames tributaries and other marshy areas of the south east of England.
One thing to consider carefully before venturing out over deep mud is that even with splatchers on, if you fall over it can be very difficult/well nigh impossible to get back onto your feet again because the mud does not provide any support as you try to push yourself up. (Thinks: time to invent 'hand splatchers' or 'splatcher gloves'
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