Consider a 2-period binomial tree where the derivative price is $f$ and the stock price is $S$. Also, let the bond be deterministic with continuous growth rate $r$ and initial value $B_0$. binomial tree
Recall the replicating strategy is at each time $t_i$ hold $\phi_i = \frac{f_{i+1}^{up} - f_{i+1}^{down}}{S_{i+1}^{up} - S_{i+1}^{down}}$ units of the stock and $\psi_i = B_0^{-1} e^{-r(i+1)\Delta t}(f_{i+1}^{up} - \phi_i S_{i+1}^{up})$ units of the bond. In particular, the value of the portfolio at time $0$ is $V_0 = \phi_0 S_0 + \psi_0 B_0$. When we arrive at time tick 1, lets say our stock price went up to $S_3$. Before rebalancing, our portfolio is worth $V_0|_{end} = \phi_0 S_3 + \psi_0 B_0e^{r \Delta t}$, and after rebalancing it is $V_1 = \phi_1 S_3 + \psi_1 B_0e^{r \Delta t}$. In order for this to be self-financing, we must have $V_1 - V_0|_{end} = 0$. However, \begin{align*} V_1 - V_0|_{end} & = (\phi_1 - \phi_0)S_3 + (\psi_1 - \psi_0)B_0e^{r \Delta t} \\ & = (\phi_1 - \phi_0)S_3 + \left(B_0^{-1} e^{-2r\Delta t}(f_7 - \phi_1 S_{7}) - B_0^{-1} e^{-r\Delta t}(f_{3} - \phi_0 S_{3})\right)B_0e^{r \Delta t} \\ & = (\phi_1 - \phi_0)S_3 + e^{-r\Delta t}(f_7 - \phi_1 S_{7}) - (f_{3} - \phi_0 S_{3}) \\ & = \phi_1 S_3 + e^{-r\Delta t}(f_7 - \phi_1 S_{7}) - f_{3} \\ & = \frac{f_{7} - f_{6}}{S_7 - S_6} S_3 + e^{-r\Delta t}(f_7 - \frac{f_{7} - f_{6}}{S_7 - S_6} S_{7}) - f_{3} \\ & = \frac{1}{S_7 - S_6} \left((f_{7} - f_{6})S_3 + (S_7 - S_6)e^{-r\Delta t}f_7 - e^{-r\Delta t}(f_{7} - f_{6}) S_{7} - (S_7 - S_6)f_{3} \right)\\ & = \frac{1}{S_7 - S_6} \left((f_{7} - f_{6})S_3 - S_6e^{-r\Delta t}f_7 + e^{-r\Delta t}f_{6} S_{7} - (S_7 - S_6)f_{3} \right)\\ & \neq 0. \end{align*}
It seems a lot of effort is put into self-financing strategies, and in fact the binomial representation theorem is used to prove the existence of them in the binomial model. Am I missing something?