How do we know that the Emancipation episode falls into the events of Captain America: Civil War?

At the start of the episode there is a TV showing a new broadcast which mentions the Sokovia Accords (and the Cap vs Tony fight).

This episode clearly takes place after Civil War so there are no common events as such. Rather AoS deals with the aftermath and the effect of the Sokovia Accords now being the "law of the land" and how that affects their operations and personnel.

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General Talbot arrives on the orders of the President as they suspect that S.H.I.E.L.D. "may have some undocumented enhanced assets working for you"....which, if course, they do.

Talbot suggests that it's time for S.H.I.E.L.D. to re-legitimise and come in from the cold but Coulson suggest that the quid pro quo would be revealing and registering any Inhumans they may have.

Coulson naturally resists since not only does he disagree with the policy on moral grounds but he needs these Inhumans to combat the threat Hive represents.

Coulson:

Nothing good ever comes from putting people on lists! Eventually they get misused or fall into the wrong hands and then innocent people suffer.

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It should, perhaps, be noted that Marvel Television Universe (if you will) acknowledges the events of the Marvel Cinematic Universe (the movies) but, for reasons that have been covered extensively elsewhere, the reverse is not true.

Events in the MTU are not referenced or acknowledged in the MCU...at least so far.