Run a tab is an informal phrase that means to keep track of a running total of money owed. It is often used in the context of a bar or restaurant, where customers are allowed to run a tab and pay for their drinks at the end of the night. The crossword solution for this term is OWE, which is an acronym for 'On Your End'. This phrase is used to indicate that the customer is responsible for paying the tab at the end of the night.