Rice?

Beyond that,
What is the bed filled with?
How deep is it?
Can you choose a different site?
What is your irrigation system?
And how often do you run it?
What is the underlying soil? Clay or?
Do you have a high water table?
Do you have a seasonal stream?

celery is one.

Rose Gold potatoes can handle a mucky area.

Celery as mentioned, cranberries, watercress, taro, water chestnuts, some melon varieties. But the ideal solution would be to fix the bed instead so it could have a more practical use.

Dave