Methods and systems are provided for operating a split exhaust engine system that provides blowthrough air and exhaust gas recirculation to an intake passage via a first exhaust manifold and exhaust gas to an exhaust passage via a second exhaust manifold. In one example, in response to a request to shut down the split exhaust engine system, an intake throttle may be closed and a first valve disposed in a secondary flow passage coupled between the intake manifold, downstream of the intake throttle, and a first exhaust manifold coupled to a first set of exhaust valves, may be opened. As a result, unburned hydrocarbons may be routed to a catalyst disposed in the exhaust passage.