<p>This article establishes local strong well-posedness and global strong well-posedness close to constant equilibria of a model coupling the primitive equations of ocean and atmosphere dynamics with Hibler’s viscous-plastic sea ice model. In order to treat the coupling conditions, an approach involving the hydrostatic Dirichlet and Dirichlet-to-Neumann operator is developed. Mapping properties of the latter operators are investigated for the first time and are of central importance for showing that the operator associated with the linearized coupled system admits a bounded <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {H}^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-calculus on suitable <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textrm{L}^q\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>L</mtext> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>-spaces. Quasilinear methods allow then to obtain the strong well-posedness results described above.</p>