Option 1: The statement that the soils of tropical rainforests are infertile is incorrect. In fact, tropical rainforests have some of the most fertile soils in the world due to the constant decomposition of organic matter.

Option 2: The statement that the vegetation in tropical rainforests is evergreen, enabling photosynthesis to take place year-round, is correct. The high amount of rainfall and consistent temperatures in these regions allow for continuous plant growth.

Option 3: The statement that tropical rainforests have been described as `deserts covered by trees` is correct. Despite the abundance of vegetation, the competition for resources in this ecosystem can be intense, leading to limited availability of nutrients and sunlight in certain areas.

Option 4: The statement that tropical rainforests are the most productive land-based ecosystem is correct. These forests have a high level of biodiversity and support a wide range of plant and animal species, making them highly productive in terms of biomass and ecological functions.

In conclusion, the correct statement about tropical rainforests is option 4: 1, 2, 3, and 4.