Suppose I have an $A_{\infty}$-space $X$, such that its unit is only a unit up to homotopy. When the space is well-behaved (well-pointed? What is the weakest condition possible?), I can replace it with a homotopy equivalent version of $X$ that has an honest unit. I read the definition for the classifying space of an $A_{\infty}$-space in Stasheff's papers. He uses honest units. Is it possible to circumvent this somehow?

Is there a functorial definition of the classifying space $BX$ that does not strictify the homotopy unit?