The standard technique is to solve the optimality conditions as a
linear system of equations. Let us rewrite your problem into standard
form
$$
\begin{align*}
\text{maximize } & -c^{T}x\text{ s.t. }\begin{bmatrix}-c_{i}^{T}\\
-A_{i}
\end{bmatrix}x+\begin{bmatrix}t_{i}\\
z_{i}
\end{bmatrix}=\begin{bmatrix}d_{i}\\
b_{i}
\end{bmatrix},\;\begin{bmatrix}t_{i}\\
z_{i}
\end{bmatrix}\in\mathrm{SOC},\\
\text{minimize } & \sum_{i=1}^{m}\begin{bmatrix}d_{i}\\
b_{i}
\end{bmatrix}^{T}\begin{bmatrix}\lambda_{i}\\
u_{i}
\end{bmatrix}\text{ s.t. }\sum_{i=1}^{m}\begin{bmatrix}-c_{i}^{T}\\
-A_{i}
\end{bmatrix}^{T}\begin{bmatrix}\lambda_{i}\\
u_{i}
\end{bmatrix}=-c,\;\begin{bmatrix}\lambda_{i}\\
u_{i}
\end{bmatrix}\in\mathrm{SOC}.
\end{align*}
$$
The classical optimality conditions are written
$$
\begin{align*}
\sum_{i=1}^{m}\begin{bmatrix}-c_{i}^{T}\\
-A_{i}
\end{bmatrix}^{T}\begin{bmatrix}\lambda_{i}\\
u_{i}
\end{bmatrix} & =-c,\\
\begin{bmatrix}-c_{i}^{T}\\
-A_{i}
\end{bmatrix}x+\begin{bmatrix}t_{i}\\
z_{i}
\end{bmatrix} & =\begin{bmatrix}b_{i}\\
d_{i}
\end{bmatrix}\;\forall i\in\{1,\ldots,m\},\\
\begin{bmatrix}\lambda_{i} & u_{i}^{T}\\
u_{i} & \lambda_{i}I
\end{bmatrix}\begin{bmatrix}t_{i}\\
z_{i}
\end{bmatrix} & =0\qquad\forall i\in\{1,\ldots,m\},
\end{align*}
$$
see Theorem 16, Alizadeh & Goldfarb, Second-order cone programming,
Math. Program., 2001. Notice that the system of equations
is linear over $x,t_{i},z_{i}$ with $\lambda_{i},u_{i}$ fixed. Substituting
$\lambda_{i}\gets\lambda_{i}^{\star}$ and $u_{i}\gets u_{i}^{\star}$
and solving the linear equations
$$
\begin{bmatrix}\lambda_{i} & u_{i}^{T}\\
u_{i} & \lambda_{i}I
\end{bmatrix}\left(\begin{bmatrix}c_{i}^{T}\\
A_{i}
\end{bmatrix}x+\begin{bmatrix}b_{i}\\
d_{i}
\end{bmatrix}\right)=0 \qquad \forall i\in\{1,\ldots,m\}
$$
yields the primal-optimal solution $x^{\star},t_{i}^{\star},z_{i}^{\star}$, assuming that the equations are full-rank or overdetermined (given strong duality, a zero-residual solution must exist). If the equations are underdetermined, then we would have to reimpose the constraint $[t_i^\star, z_i^\star]\in \mathrm{SOC}$ and solve a SOCP feasibility problem.