Anchugearous fabricus
Still grows loosely but has a slightly more ordered structure than it's ancestor and is slightly stronger. Their stronger thicker net-like structure is covered in smaller hair-like protrusions that are softer and greener. As well as drinking up sunlight these structures produce spores.
A mass rooted to the seafloor is able to grow up to about a metre out from the seafloor before the waves and currents (cause it to move too much (due to occupying shallow sea and coast, often in close proximity to the surface) and it breaks off. Usually they will stop growing outwards before this happens, detecting the stress before it becomes too much, and instead direct growth towards expanding it's anchorage and growing more outwardly. It's anchored position in the shallows between the islands allows it consistent warmth, light and relative security to grow tall from the seafloor without being carried out to sea.
That said, some do lose their anchor. Most of the time they get carried out to sea where they can survive sub-optimally for a time. In this event they direct all of their resources towards massive quantity asexual sporing in a last effort to pass on their genes before they die. These spores become finely dispersed in the ocean and maybe only a few will reach coasts, but it sometimes works especially in Zone 5 where there are a lot of islands. If they are lucky however they can become snagged on a rock or even another of their species and re-root on whatever surface they get caught on. A. greenensis grows in large patches at the mercy of the weather, so those with more strength and order to the pattern of their structure would stand greater chance of keeping themselves from fragmenting to the point of death of the algae. The branching net-like structure could embed into the spaces between sand and rock from the spore, certainly not as reliable as a real root system but just a primitive form of anchorage. The differentiation of the branching tissue and the sporing tissue would already have existed to a lesser degree in the ancestor or if not would have arisen from mutations that were selected for and became more extreme as they give A. fabricus greater surface area for photosynthesis and the specialization of the tissues for reproduction at the expense of robustness spares tissues in other structures of that burden so energy can be spent on structural integrity. Many small islands mean there's a lot of coastline, and this gives more chance for spores and fragments to encounter the shallow and coastal seabeds and anchor. I suspect most other zones wouldn't have been as suitable a habitat for this.