We develop linesearch algorithms intended for globalization of convergence of the piecewise Levenberg–Marquardt method for constrained piecewise smooth equation. Conditions ensuring global convergence properties and asymptotic superlinear convergence rate are proposed. The peculiarities of the global convergence results in the piecewise smooth case are discussed and illustrated by examples. We also provide numerical results for unconstrained and constrained reformulations of nonlinear complementarity problems, comparing the performance of the globalized piecewise Levenberg–Marquardt algorithm with some relevant alternatives.