It is a long-standing open problem to support verified compilation of multi-threaded programs compositionally when sharing of stack data between threads is allowed. Although certain solutions exist on paper, none of them is completely formalized because of the difficulty in simultaneously enabling sharing and forbidding modification of stack memory in presence of arbitrary memory operations (e.g., pointer arithmetic). We present a compiler verification framework that solves this open problem in the setting of cooperative multi-threading. To address the challenges of sharing stack data, we introduce threaded Kripke memory relations (TKMR) to support both protection and sharing of stacks in a multi-stack memory model. We further introduce threaded forward simulations parameterized by TKMR to capture semantics preservation for compiling program modules in multi-threaded contexts. We show that threaded forward simulations are both horizontally composable— thereby enabling the compositional verification of open threads and heterogeneous modules—and vertically composable—thereby enabling composition of compiler correctness for multiple compiler passes. Furthermore, threaded forward simulations can be converted into backward simulations. We apply this framework to 18 passes of CompCert to get CompCertOC, the first optimizing verified compiler that supports compositional verification of cooperative multi-threaded programs with shared stacks.