This is one of those questions I never thought to ask. But apparently it was a big enough issue in the late 19th Century to address in the 1873 Boston Catechism (p 64)…no joke:
Q: Should we abstain from spitting after having received communion?
A: Yes; we should endeavor to refrain from it for the space of a quarter of an hour, out of respect for the sacrament.
Everybody got that? No spitting for 15 minutes after communion.