MIAMI—Russia is solidifying its footing close to U.S. shores, in the Caribbean, a region Moscow abandoned after the Cold War and has gradually returned to with investment, diplomacy and military hardware.

In recent months Russia has developed a multidimensional relationship with the tiny nation of Grenada, whose name has resonated with many Americans since President Ronald Reagan invaded in 1983 rather than see another leftist-revolutionary government follow Cuba and mature in the Caribbean.

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