A non-splitting method for tridiagonalizing complex symmetric (non-Hermitian) matrices is developed and analyzed. The main objective is to exploit the purely structural symmetry in terms of runtime performance. Based on the analytical derivation of the method, Fortran implementations of a blocked variant are developed and extensively evaluated experimentally. In particular, it is illustrated that a straightforward implementation based on the analytical derivation exhibits deficiencies in terms of numerical properties. Nevertheless, it is also shown that the blocked non-splitting method shows very promising results in terms of runtime performance. On average, a speed-up of more than three is achieved over competing methods. Although more work is needed to improve the numerical properties of the non-splitting tridiagonalization method, the runtime performance achieved with this non-unitary tridiagonalization process is very encouraging and indicates important research directions for this class of eigenproblems.