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The framework is motivated by applications in the field of obstacle-type quasi-variational inequalities and implicit obstacle problems. It is discussed in a general functional analytic setting and allows for inexact function evaluations and Newton steps. Moreover, if a certain contraction assumption holds, we show that it is possible to globalize the algorithm by means of the Banach fixed-point theorem and to ensure\n                    <jats:italic>q<\/jats:italic>\n                    -superlinear convergence to the problem solution for arbitrary starting values. By means of a localization technique, our Newton method can also be used to determine solutions of fixed-point equations that are only locally contractive and not uniquely solvable. We apply our algorithm to a quasi-variational inequality which arises in thermoforming and which not only involves the obstacle problem as a source of nonsmoothness but also a semilinear PDE containing a nondifferentiable Nemytskii operator. Our analysis is accompanied by numerical experiments that illustrate the mesh-independence and\n                    <jats:italic>q<\/jats:italic>\n                    -superlinear convergence of the developed solution algorithm.\n                  <\/jats:p>","DOI":"10.1007\/s10589-025-00722-8","type":"journal-article","created":{"date-parts":[[2025,11,10]],"date-time":"2025-11-10T13:49:49Z","timestamp":1762782589000},"page":"1-55","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A globalized inexact semismooth Newton method for nonsmooth fixed-point equations involving variational inequalities"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7616-3293","authenticated-orcid":false,"given":"Amal","family":"Alphonse","sequence":"first","affiliation":[]},{"given":"Constantin","family":"Christof","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Hinterm\u00fcller","sequence":"additional","affiliation":[]},{"given":"Ioannis P. 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