Heat the grill. Split open the bagels, lay them on a baking sheet, then toast both sides until golden. Meanwhile, tip the tuna into a bowl and add the mayonnaise, lemon juice and chives. Season to taste and mix well.
Spread the tuna mix over the bottom half of each of the 4 bagels and top with tomato slices. Sprinkle over a handful of grated cheddar, then grill for 1 min until melted. Finish with the bagel tops and serve.