Is it "toute suite" or "tout de suite"?

Sometimes you hear people use a French expression in English pronounced “toot sweet”. The meaning is like Spanish pronto or rural American English right quick. Often, when spelled out, this expression is spelled as the two French words tout suite.

But shouldn’t this expression be spelled tout de suite? IIRC, the French expression pas tout de suite means in English approximately “not so fast”. It seems pretty likely to me that pas tout de suite and tout de suite are conceptually related expressions.

BTW, for Francopone Dopers: what is the literal meaning of tout de suite in English? “All of … ?” I know this is an idiomatic expression, but still I’m wondering what the word suite translates into.