## 1 Introduction
Slip preference is a term that refers to the expected (theoretical) fault slip under a stress regime, and in the best-fit stress inversion methods, it is assumed to coincide with the direction and the sense of the maximum resolved shear stress (the Wallace-Bott hypothesis [15; 17]). In general, if the MA of each activated fault is less than \(20^{\circ}\) ([1; 34; 35] and references therein), the stress tensor solutions are considered satisfactory, though the threshold angle influences the final results of the methods [36]. However, an individual fault might exhibit a polyphase activity in regions with complex tectonic history. As a result, the recorded fault-slip data are heterogeneous (or polyphase). In such cases, it can be quite puzzling to define exactly which faults have co-functioned under the different driving stress regimes with only the MA minimization criterion. This problem remains unsolved when using either the separation of a fault/slipdataset into independent subsets (hard division) [37; 38; 39] or not (soft division) [40]. As a result, in the presence of heterogeneous fault-slip data, the estimation of the optimal stress through the application of the best-fit stress inversion methods is difficult to be reached [38] because such solutions might be misleading due to the influence of faults belonging to other phases [36; 38]. Such an estimation can be even more precarious, especially when stress tensors are calculated from \(<10\) fault-slip data, since such solutions can be found even from randomly generated faults [41]. Although the heterogeneous fault-slip data should be classified into homogeneous groups based on field evidence, this is hard to establish in most cases, like the Voltri Massif of the Ligurian Alps. Therefore, additional compatibility criteria based on the relationship between the fault activation and the driving stress regime are needed to separate the fault-slip data into homogeneous groups. Regardless of the above, the need to define the stresses drifts scientists to analyze their fault-slip data with the available open-source software, making the process and the used software progressively widely accepted. Such widely used best-fit stress inversion methods are provided by the FSA software ([42] and subsequent modifications), which uses the Monte Carlo search method [1] and Win-Tensor software [43]. The process is iterative and repeats until the remaining data cannot define a physically meaningful stress tensor. Several published articles are based on this inversion approach without using tectonost stratigraphic criteria, e.g., [29; 32]. The study in [32] examined the polyphase brittle deformation in Voltri Massif and suggested a paleostress history using the FSA software [42] to separate heterogeneous fault-slip data into homogeneous groups, and the Win-Tensor software [43] for verifying this grouping. Both software use best-fit stress inversion algorithms. On the other hand, the study in [44], based not only on the MA minimization criterion of the best-fit stress inversion method, but also on additional compatibility criteria induced by the Slip Preference Analysis concerning the fault activations, developed a new separation and stress inversion method, the Tensor Ratio Method (TRM). This fact motivated us to test and examine whether additional compatibility criteria could assist with data separation and provide better resolved stress tensors that can enhance our understanding of polydeformed regions as the Voltri study area. The purpose of this study is to examine: (1) whether the results concerning the fault activations in the region and the paleostress history deviate or not from that of [32], (2) the efficiency of the differently applied stress inversion methods and the usefulness of implementing different constraining compatibility criteria among the fault-slip data, and (3) a better understanding of the stress regimes as provided by the site and bulk resolved stress tensors.