This paper considers the global classical solvability (well-posedness) of a mixed initial-boundary value problem for semilinear hyperbolic systems with nonlinear reaction coupling of Lotka-Volterra type. The reaction nonlinearity is not globally Lipschitz in L-2 and has Lipschitz properties depending on an L-infinity-norm bound. The well-posedness problem is reformulated in an abstract setting as a modified Cauchy problem with homogeneous boundary conditions and solved based on the Banach contraction mapping theorem. Extra regularity of the local solutions in Sobolev spaces is shown based on Moser-type inequalities. It is shown that global existence of classical solutions holds if a uniform a priori bound on the L-infinity-norm of the solution and boundary term exists.