Connections compatible with degenerate metric structures are known to possess peculiar features: on the one hand, the compatibility conditions involve restrictions on the torsion; on the other hand, torsionfree compatible connections are not unique, the arbitrariness being encoded in a tensor field whose type depends on the metric structure. Nonrelativistic structures typically fall under this scheme, the paradigmatic example being a contravariant degenerate metric whose kernel is spanned by a one-form. Torsionfree compatible (i.e. Galilean) connections are characterised by the gift of a two-form (the force field). Whenever the two-form is closed, the connection is said Newtonian. Such a nonrelativistic spacetime is known to admit an ambient description as the orbit space of a gravitational wave with parallel rays. The leaves of the null foliation are endowed with a nonrelativistic structure dual to the Newtonian one, dubbed Carrollian spacetime. We propose a generalisation of this unifying framework by introducing a new non-Lorentzian ambient metric structure of which we study the geometry. We characterise the space of (torsional) connections preserving such a metric structure which is shown to project to (resp. embed) the most general class of (torsional) Galilean (resp. Carrollian) connections.