We present a 3D contrast source inversion scheme for electromagnetic data in conductive media. We consider only contrasts in electric conductivity but allow the medium to be transversely isotropic in the vertical direction. This has applications in, for instance, inversion of marine controlled-source electromagnetic data. The contrast source inversion (CSI) method is based on the integral equation formulation of electromagnetic field propagation and solves the inverse problem of determining the conductivity structure of the subsurface. The method minimizes a cost functional that enforces both data fidelity and that the solution satisfy the Lippmann-Schwinger equation. Further regularization is introduced linearly into the cost functional to incorporate prior model information. Although the problem is nonlinear, the chosen strategy splits the minimization problem into two linear problems, which are solved alternatingly. To this end, contrast sources are introduced, which may be interpreted as sources emitting the scattered field from a scattering object. Two synthetic and two real field examples are inverted, which demonstrates the method and how the transversely isotropic in the vertical direction (TIV) inversion performs compared with isotropic inversion. The CSI method is found to be applicable to real field examples, and the results show that a TIV inversion is preferred over isotropic to identify weak anomalies in these examples. The reason for this is that both the horizontal and vertical conductivity affects the signal propagation in the overburden.