In a Tactical Field Care situation, what is the first step for a casualty lying on his back with an entrance wound in the chest?

In a Tactical Field Care situation where a casualty is lying on their back with an entrance wound in the chest, the first step should be to use a vented chest seal to cover the wound. This action is critical because an entrance wound in the chest can potentially lead to a life-threatening condition known as a tension pneumothorax, especially if the wound allows air to enter the pleural space during inhalation and prevents it from escaping during exhalation. Using a vented chest seal helps to create a one-way valve that allows air to escape from the pleural cavity while preventing additional air from entering, thereby stabilizing the casualty's respiratory status.

While assessing for a head injury and monitoring the casualty’s breathing are important steps in the overall evaluation and management of trauma, they are not immediate priorities compared to addressing an open chest wound. Additionally, applying a tourniquet is appropriate for controlling significant extremity hemorrhage and would not be indicated as the first action for a chest wound. Addressing the chest wound with a vented seal directly tackles the most immediate threat to life in this scenario.