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A general framework for the development of high-order compact schemes has been proposed recently. The core steps of the schemes are composed of the following. 1). Based on a kinetic model equation, from a generalized initial distribution of flow variables construct a time-accurate evolution solution of gas distribution function at a cell interface ; 2). Introduce the WENO-type weighting functions into the time-derivative of the cell interface flux function in the multistage multi-derivative time stepping scheme to cope with the possible impingement of a shock wave on a cell interface within a time step; 3). Take moments of interface gas distribution function to obtain the time-accurate flow variables and the corresponding fluxes at the cell interface, and update the cell-averaged flow variables and their gradients inside each control volume; 4). Within the physical domain of dependence of the reconstructed cell, based on the cell-averaged flow variables and their gradients develop compact initial data reconstruction to get initial flow distributions at the beginning of next time step. A compact gas-kinetic scheme (GKS) up to sixth-order accuracy in space and fourth-order in time has been constructed on 2D unstructured mesh before. In this paper, the compact GKS up to fourth-order accuracy on 3D tetrahedral mesh will be further constructed with the focus on the WENO-type initial data reconstruction. Nonlinear weights are designed to achieve high-order accuracy for the smooth Navier-Stokes solution and keep super robustness in 3D computation with strong shock interactions. The fourth-order compact GKS can use a large time step with CFL number $0.6$ in the simulations from subsonic to hypersonic flow. A series of test cases are used to validate the scheme. The high-order compact GKS is ready for 3D applications with complex geometry.