Session types, types for structuring communication between endpoints in distributed systems, arerecently being integrated into mainstream programming languages. In practice, a very importantnotion for dealing with such types is that of subtyping, since it allows for typing larger classes ofsystem, where a program has not precisely the expected behavior but a similar one. Unfortunately,recent work has shown that subtyping for session types in an asynchronous setting is undecidable.To cope with this negative result, the only approaches we are aware of either restrict the syntaxof session types or limit communication (by considering forms of bounded asynchrony). Bothapproaches are too restrictive in practice, hence we proceed differently by presenting an algorithm forchecking subtyping which is sound, but not complete (in some cases it terminates without returninga decisive verdict). The algorithm is based on a tree representation of the coinductive definition ofasynchronous subtyping; this tree could be infinite, and the algorithm checks for the presence offinite witnesses of infinite successful subtrees. Furthermore, we provide a tool that implements ouralgorithm and we apply it to many examples that cannot be managed with the previous approaches.