Image segmentation and registration are pivotal preliminary steps in image analysis exhibiting a dual nature, especially in a multimodal context where salient features are to be matched. As such, intertwining them in a unified framework reduces uncertainty propagation and yields positive mutual influence. Registration compensates for weak boundary definition and encodes intrinsic topological requirements like preserving contextual relations between objects. In return, accurate segmented structures foster relevant registration, breeding reliable estimations of the deformation pairing the encoded structures. These observations underpin the proposed contribution blending variational techniques (versatility/adaptability/interpretability) and deep-learning-based approaches (more proficient at handling computationally intensive tasks) through coordinate Multi-Layer Perceptron (MLP) with periodic sinusoidal activation functions. More precisely, in a hybrid nonlinear-elasticity-grounded framework, an unsupervised pairwise joint segmentation/registration 2D model is introduced. Working in the class of mappings with finite distortion enables one to guarantee that the engendered deformation is a homeomorphism. Also, combining a directional-total-variation-based term promoting gradient alignment with the segmentation task, viewed as a substitute for classical intensity-based data terms in registration, enlarges the scope of applications to multimodal images. The existence of a minimiser for the optimisation problem constitutes the core of the paper. Eventually, we test and evaluate our model on both synthetic and medical images to exhibit the accuracy and relevance of our model.