High-resolution, multi-wavelength, and time-domain observations of the Galactic centre black hole candidate, Sgr A*, allow for a direct test of contemporary accretion theory. To date, all models have assumed alignment between the accretion disc and black hole angular momentum axes, but this is unjustified for geometrically thick accretion flows like that onto Sgr A*. Instead, we calculate images and spectra from a set of simulations of accretion flows misaligned ('tilted') by 15 degrees from the black hole spin axis and compare them with millimetre (mm) to near-infrared (NIR) observations. Non-axisymmetric standing shocks from eccentric fluid orbits dominate the emission, leading to a wide range of possible image morphologies. These effects invalidate previous parameter estimates from model fitting, including estimates of the dimensionless black hole spin, except possibly at low values of spin or tilt. At 1.3mm, the images have crescent morphologies, and the black hole shadow may still be accessible to future mm-VLBI observations. Shock heating leads to high energy electrons (T > 10^12 K), which can naturally produce the observed NIR flux, spectral index, and rapid variability ('flaring'). This NIR emission is uncorrelated with that in the mm, which also agrees with observations. These are the first models to self-consistently explain the time-variable mm to NIR emission of Sgr A*. Predictions of the model include significant structural changes observable with mm-VLBI on both the dynamical (hour) and Lense-Thirring precession (day-year) timescales; and ~30-50 microarcsecond changes in centroid position from extreme gravitational lensing events during NIR flares, detectable with the future VLT instrument GRAVITY. If the observed NIR emission is caused by shock heating in a tilted accretion disc, then the Galactic centre black hole has a positive, non-zero spin parameter (a > 0).