I use regular shitty brown cardboard in my otherwise neglected garden beds, it will eventually decompose and get eaten by the wormies, but lasts longer than newspaper.
My wife made us a gothic-arch greenhouse a few years back, I grew cherry tomatoes in it. The soil definitely had to be covered for a season to kill everything first. We also tilled in some horseshit from my friend's farm, so that had to decompose for a bit.
After that we realized that even though we had no weeds, humidity was off the charts as @cannabissequoia mentioned. So we used those expensive silver tarps on everything except the few spots where the tomatoes came up, and openings for watering.
This way we were able to actually close up the greenhouse to keep some heat and not have everything rot inside from otherwise insane humidity. Keep in mind though this was around Vancouver, BC, and it is pretty damn wet here. Rainforest wet.