Happily spending my holidays in the Fiji Islands, I noticed a couple of days ago that when the moon was very high in the sky (my shadow was maybe 0.5-1m long) the tide was low. That really puzzled me. How does this happen?
This is based on the overly-simplified model of tides being the result of tidal bulges. As I explained in my answer to a related question on the physics.SE sister site, those tidal bulges do not and cannot exist.
Instead, the tides are dynamic responses to the tidal forcing functions from the Moon and the Sun, with the orbits of the three bodies about one another, the Earth's rotation rate, and the geometry of the oceanic basins all playing roles in the dynamic response. For any point on the surface of the Earth's oceans, the tides can be described in terms of a composition of frequency responses to those cyclic forcing functions. See the related question Definitions of tidal harmonic constituents…? for details.
At many places, Fiji included, the key component of the tides is the $M2$ tidal constituent, with a period of 12.42 hours. This is the principal response to the tides induced by the Moon. If Newton's equilibrium (tidal bulge) theory of tides was correct, every oceanic place on the Earth's surface would see high tide when the Moon is at zenith and nadir, and low tides when the Moon is at the horizon. Except for a few places, this is not what is observed. In Fiji, the situation is nearly reversed, as depicted by the tide chart below for Nadi on the west coast of Viti Levu, where high tides are more or less in sync with moonrise and moonset.
Source: https://www.tide-forecast.com/locations/Nandi-Waters-Viti-Levu-Fiji-Islands/tides/latest, accessed 2 November, 2017.
Laplace's dynamic theory of the tides says that the oceans' response to each of the tidal constituents comprises a set of amphidromic systems. Each such system is centered about an amphidromic point, a place with a null response to the component in question. A wave whose amplitude increases with distance from the the amphidromic point rotates about the amphidromic point at the constituent frequency. The $M2$ tidal response is depicted below. The $M2$ amphidromic points are at the centers of the dark blue areas. The white lines emanating from the amphidromic points are cotidal lines, curves along which high tide occurs at the same time. The different colors indicate height, which again increase with distance from the governing amphidromic point.