The ability of the human brain to build complex expressions from simpler parts is fascinating, but the ability of the bilingual brain to do so is perhaps even more remarkable. When highly proficient bilinguals converse, they can fluidly switch from one language to another even inside sentences. Thus, they build expressions using words from more than one language. How are bilinguals able to compose words across different languages in real time? While robust evidence has implicated the left anterior temporal lobe (LATL) for the composition of words within one language, we do not know how the LATL, or other regions implicated for composition, operates when the language switches. We also do not know whether prefrontal regions associated with language control are recruited for language switching during composition. We addressed these questions with magnetoencephalography measurements in bilinguals who are fluent in two typologically distant languages, English and Korean. We observed early composition effects in the LATL at ∼200 ms that were unaffected by either language or orthography switching, which was also varied (Hangul vs Roman alphabet). Thus, the combinatory mechanism at 200 ms housed in the anterior temporal cortex appears blind to the language in which its input concepts are expressed. However, in later time windows, language and orthography switching interacted both in regions implicated for composition [LATL, ventromedial prefrontal cortex, left inferior frontal gyrus (LIFG)] as well as in regions associated with language control (ACC, LIFG). This establishes a starting point for understanding how bilingual brains code switch: words are initially combined without consideration of which language they come from, but language switching affects later processing.