There are lots of variations worldwide, and many different cultures have used "mud" for houses.

In the "settler days" in the US prairie states, it was more common to use "sod" for the most primitive houses. In places where there were few trees, you could cut out slabs of sod, and the grass roots would hold it together to be piled like bricks. Walls typically were 6 to 10 inches thick - the width of the slabs, and once the sod was "off the ground" most water ran off and the roots that held the slabs together dried into rather tough "strings" that resisted rotting out and continued to hold the sod together - sometimes for many decades.

In places where there were "hills" the "dugout" was also used quite a lot. You'd dig a "pit silo" into the side of a hill, and put sod walls onto the front.

Most of the settlers looked to wood for roof rafters, but the covering there was often also sod. I haven't seen any relics of either kind with "roof arches" as would be needed to make a roof with no timbers.

In some areas (not very commonly in the US) the mud houses are actually more of a "wattle" construction, with small limbs/twigs woven together, and the "mud" is mixed or rubbed in - like a mortar - just to close the gaps. Sometimes straw (any dead grassy stuff) is mixed with "mud" in the wattle construction or in "mostly just mud" houses, but in both cases the preferred "mud" often is animal waste from whatever large domestic animals are conveniently available. (It dries harder, and once hard is very water-resistant. It also fairly consistently has undigested "fibers" that help strengthen the mortar.)

Adobe is generally a little more durable, but it requires a fairly decent access to enough water to make a paste of the local soil in order to form the "bricks," and it's only really successful if the soil used has fairly high clay content. Fortunately the two requirements are "compatible" since in any place that has free running water, even if only seasonally, there's usually enough clay to be found. And straw fairly often was/is used mixed into the adobe bricks for a bit of additional strength.

The better durability of adobe becomes questionable in wetter climates, and a well-built sod house may stand up better where there's more than 40-60(?) inches per year of rainfall. Good drainage helps for either.

John