In order to maximize the signal-to-interference-plus-noise ratio (SINR) under a constant-envelope (CE) constraint, a fast and efficient joint design of the transmit waveform and the receive filter for colocated multiple-input multiple-output (MIMO) radars is essential. Conventional joint optimization is performed using nonlinear optimization techniques such as the semidefinite relaxation (SDR) algorithm. In this letter, we propose a novel manifold-based alternating optimization (MAO) method, which reformulates the waveform optimization subproblem as an unconstrained optimization problem on a Riemannian manifold. We present the geometrical structure of the feasible region and derive the explicit expressions for the Riemannian gradient and the Riemannian Hessian, thus the reformulated optimization could be solved by using the Riemannian trust-region (RTR) algorithm. Numerical experiments demonstrate that the proposed method has faster convergence with reduced computational cost compared with conventional SDR-based algorithm in Euclidean space.