MSL uses the forecast’s global mean sea level. LAT uses the lowest predicted astronomical tide under average weather at the offshore forecast cell. A low tide can sit above that zero. Wind and pressure also affect the water-level forecast, so providers can differ even when both use LAT.
Modeled LAT from Poncelet, Billant & Saunier (FES2014). Contains modified Copernicus Marine Service information. LAT grid · CC BY 4.0 · Copernicus local mean sea level.