Local corner smoothing method is commonly adopted to smooth linear (G01) tool path segments in computer numerical control (CNC) machining to realize continuous motion at transition corners. However, because of the highly non-linear relation between the arc-length and the spline parameter, and the challenge to synchronize the tool tip position and tool orientation, real-time and high-order continuous five-axis tool path smoothing and interpolation algorithms have not been well studied. This paper proposes a real-time C3 continuous corner smoothing and interpolation algorithm for five-axis machine tools. The transition corners of the tool tip position and tool orientation are analytically smoothed in the workpiece coordinate system (WCS) and the machine coordinate system (MCS) by C3 continuous PH splines, respectively. The maximum deviation errors of the smoothed tool tip position and the tool orientation are both constrained in the WCS. An analytical synchronization algorithm is developed to guarantee the motion variance of the smoothed tool orientation related to the tool tip displacement is also C3 continuous. The corresponding real-time interpolation method is developed with a continuous and peak-constrained jerk. Simulation results verify that the maximum deviation errors caused by the tool path smoothing algorithm are constrained, and continuous acceleration and jerk of each axis are achieved along the entire tool path. Comparisons demonstrate that the proposed algorithms achieve lower amplitude and variance of acceleration and jerk when compared with existing methods. Experiments show that the proposed five-axis corner smoothing and interpolation algorithms are serially executed in real-time with 0.5-ms cycle.