Asynchronous programming models (APM) are gaining more and more traction, allowing applications to expose the available concurrency to a runtime system tasked with coordinating the execution. While MPI has long provided support for multi-threaded communication and non-blocking operations, it falls short of adequately supporting APMs as correctly and efficiently handling MPI communication in different models is still a challenge. Meanwhile, new low-level implementations of light-weight, cooperatively scheduled execution contexts (fibers, aka user-level threads (ULT)) are meant to serve as a basis for higher-level APMs and their integration in MPI implementations has been proposed as a replacement for traditional POSIX thread support to alleviate these challenges. In this paper, we first establish a taxonomy in an attempt to clearly distinguish different concepts in the parallel software stack. We argue that the proposed tight integration of fiber implementations with MPI is neither warranted nor beneficial and instead is detrimental to the goal of MPI being a portable communication abstraction. We propose MPI Continuations as an extension to the MPI standard to provide callback-based notifications on completed operations, leading to a clear separation of concerns by providing a loose coupling mechanism between MPI and APMs. We show that this interface is flexible and interacts well with different APMs, namely OpenMP detached tasks, OmpSs-2, and Argobots.