Imagine we have two thermodynamic systems, one a mass of ice and the other an equal mass of liquid water, with both at 273.16K. Each system is isolated, except that they can interact with each other across a boundary that permits the exchange of heat but not matter or work.
What will the two systems look like at equilibrium? Somehow I want to automatically imagine that each system will be identical, a combination of liquid water, ice, and water vapor at 273.16K. But if this is true then the two systems were initially at the same temperature but not in thermodynamic equilibrium, an apparent violation of the zeroth law of thermodynamics.