When the government runs a budget deficit, what must it eventually do in

order to pay back its debt?

Have a budget surplus.

The government has two options, with regards to paying back its debts; taxation and open market operations. Through taxation, tax rates per unit may be increased, which subsequently raises enough money to be used to pay up the debt. As for open market operations, the government, through its treasury or exchequer may issue risk free treasury bonds and bills to members of the public at fixed coupon rates. Whatever funds raised from the bond and bill sale are eligible for repaying the debt.