Artificial spin ices (ASIs) are arrays of interacting nanomagnets in which phase transitions can be imaged and controlled. Studies of the thermodynamic have focused mainly on the square and kagome lattices, and on their variants. Here we study artificial spin ice on the CaVO lattice, formed of squares and octagons. At fixed lattice constant, the ratio r of the square and octagon link lengths tunes the interactions strongly while changing the geometry only slightly. Magnetic force microscopy and Monte Carlo simulations show a continuous bridge from square-ice-like to kagome-like behavior. At small r the system has a single ordering transition, whereas at large r it passes from a correlated paramagnet to charge order and then long-range order. The two regimes meet in a multicritical region, accurately modeled by a Blume-Capel model of the octagon toroidal moments. From this picture, we design a comparative magnetometry protocol that unambiguously separates interaction effects from single-nanomagnet effects in thermally active samples. Finally, the charge-ordered region maps onto a tunable dimer model, with a Coulomb phase and pinch points along the line between two ordered dimer phases. The CaVO ASI thus provides access to a rich phase diagram only through geometric tuning, without requiring extreme experimental conditions.