Answer: "The opposite side of the coin" is an idiomatic expression that refers to the other aspect or the different perspective of a situation or issue. It means that there are two sides to a story or an issue, and while one aspect may be apparent, there is also another side that should be considered.
The phrase is often used to indicate that there are both positive and negative aspects to a situation, and that both sides should be taken into account. For example, one could say, "The new project will bring many benefits, but the opposite side of the coin is that it will require a significant investment of time and money."
In essence, the phrase "opposite side of the coin" emphasizes the idea that things are not always as simple as they seem and that it is important to look at the different angles of a situation before coming to a conclusion or making a decision.
Explanation: In this particular context, the phrase "the opposite side of the coin" is used to refer to the negative consequence or the other aspect of a situation. The previous sentence states that Army intelligence was sharing decrypted and translated messages with the British through an arrangement approved by the President. The sentence following the phrase in question, "considerable embarrassment to Anglo-American relations when the British got wind of the Hull-Nomura talks", indicates that this sharing of information had negative consequences as it caused embarrassment in the relationship between the US and the UK when the British found out about the Hull-Nomura talks through the decrypted messages. Therefore, the phrase "the opposite side of the coin" in this context is highlighting the negative aspect of the situation.