The way to improve sandy soil is to add clay. You can substitute organics and make it work for a time, but you're always going to be adding organics and struggling to keep up with the constant decay. Compost and pretty much anything plant-like is mostly cellulose (C6 H10 O5), which decays to CO2 and H2O. What few minerals that are contained in plant-life will just wash away with the rains since there is nothing for the minerals to adsorb to in the sand. The decay is fast in sand because of the high amount of oxygen that can reach the organic matter contained in sand. The amount of humus left is fairly small compared with the giant pile of compost you started with... and then, what holds the humus in the sand against the rains trying to leach it away?
Biochar is good, but consider how much of that you have to make. Then you have to load it with minerals. Biochar is an amendment to soil that is already decent.
Plants need Ca, Mg, K, HPO4, NO3, NH4, SO4, B, Fe, Si, Cl, Mn, Zn, Cu, MoO4, Ni, Se, Na, and air, and water. All of these elements need to be in the elemental form, atoms (except HPO4, NO3, NH4,SO4, and MoO4, which are simple compounds). Clay has a surface area 17,600 times more than silt and 400,000 times more than sand. Source. It's that surface area that retains water (water clings to surfaces) and supplies lots of places for ions to stick.
An ideal soil is composed of 45 percent mineral (sand, silt and clay), five percent organic material (humus or plant debris and soil organisms), 25 percent water and 25 percent air.
Of the 45% that is sand, silt, and clay, the mix should be 40% sand, 40% silt, 20% clay.
A must read: It describes everything you want to know about soil and more.
The most influential factor in stabilizing soil fertility are the soil colloidal particles, clay and humus, which behave as repositories of nutrients and moisture and act to buffer the variations of soil solution ions and moisture.
A colloid is a small, insoluble, nondiffusible particle larger than a molecule but small enough to remain suspended in a fluid medium without settling. Most soils contain organic colloidal particles called humus as well as the inorganic colloidal particles of clays. The very high specific surface area of colloids and their net negative charges, gives soil its great ability to hold and release cations in what is referred to as cation exchange
Humus is the penultimate state of decomposition of organic matter; while it may linger for a thousand years, on the larger scale of the age of the other soil components, it is temporary. It is composed of the very stable lignins (30%) and complex sugars (polyuronides, 30%), proteins (30%), waxes, and fats. Its chemical assay is 60% carbon, 5% nitrogen, some oxygen and the remainder hydrogen, sulfur, and phosphorus. On a dry weight basis, the CEC of humus is many times greater than that of clay.
Most of the soil's CEC occurs on clay and humus colloids, and the lack of those in hot, humid, wet climates, due to leaching and decomposition respectively, explains the relative sterility of tropical soils. Live plant roots also have some CEC.
If I found myself on a tropical island wanting to grow vegetables, I would dig down below the sand in search of clay to add to my garden soil. I would search for hardwood trees which manufacture protein and have lots of lignin to make humus. Of course, I would take what wood I could find, chip it, and add it to a compost pile and the garden soil. If I found grass growing anywhere on the island, I would harvest that soil for my garden (grass is high in protein and takes good soil to grow it). I would construct something to limit rainfall to 1/2 inch a week (0.1 inches per day) on the soil. Any more than 0.1 - 0.2 inches per day will result in leached nutrients, destroying all my hard work. I would gather any source of calcium I could find... seashells, bones, fish (calcium is utmost importance). Anything green (leaves, seaweed, etc) goes into the compost pile and garden soil as a source of nitrogen. I would make biochar and add it to my compost pile first, then to my garden soil. Any ash from any fires would be gathered into the garden soil (ash from hardwood trees is 30% Ca, 10-15% K, 7% Mg. Source.
I think it can be done. The first step is knowing exactly what you need to do and why. The rest is just a matter of doing it.
It might be easier to buy clay and water soluble fertilizer to supply all the nutrients in proper ratios for each specific plant. Calcium, magnesium, and potassium will displace sodium.
Al3+ replaces H+ replaces Ca2+ replaces Mg2+ replaces K+ same as NH4+ replaces Na+