The pilot reported that he was conducting practice takeoffs and full-stop landings with the airplane owner in the right seat for the purposes of achieving tailwheel airplane currency. The airplane insurance company required pilots to be checked out by the airplane owner before naming them on the policy. The airplane owner was not a flight instructor. The pilot stated that on the second landing, the airplane touched down on the runway centerline and on all three wheels simultaneously. As the airplane slowed, it began to veer to the left. The pilot responded with right rudder input. The airplane subsequently veered to the right "more assertively" than he had intended. Rather than risk striking a runway light while attempting to keep the airplane on the runway, he elected to enter what appeared to be an open grass area adjacent to the runway pavement. However, about 25 feet into the grass, the airplane encountered "marshy area" with standing water and soft mud, causing it to nose over. The airplane sustained substantial damage to the vertical stabilizer and fuselage. The pilot stated that there were no failures or malfunctions associated with the airplane before the accident. He commented that he had received a tailwheel airplane endorsement about 3 years before the accident, with limited tailwheel airplane time accumulated since then. He added that the airplane owner suggested that he be more "assertive with the rudders" after the pilot's initial landing during the accident flight.