Let $(B,\mathscr{P})$ be an integral tropical manifold. Then a multi-valued piecewise linear function $\varphi$ on $B$ is a collection of continuous functions on an open cover $\{(U_{i},\varphi_{i})\}$ such that $\varphi_{i}$ is affine linear on each cell of $\mathscr{P}$ intersecting $U_{i}$, and on $U_{i}\cap U_{j}$, $\varphi_{i}-\varphi_{j}$ is affine linear. Furthermore, for any $\tau\in\mathscr{P}$, let $S_{\tau}:U_{\tau}\rightarrow\mathbb{R}^{k}$ be the induced fan structure. Then there is a piecewise linear function $\varphi_{\tau}$ on the fan $\Sigma_{\tau}$ such that on $U_{i}\cap U_{\tau}$, $\varphi_{i}-\varphi_{\tau}\circ S_{\tau}$ is affine linear. Here we will always assume that each linear part of $\varphi_{i}$ has differential in $\check{\Lambda}$, i.e., $\varphi_{i}$ has integral slopes.