Summary: Let \(F\) be a number field with ring of adeles \(\mathbb{A}\) and let \(K/F\) be a quadratic extension. We prove part of a relative trace identity between \(\mathrm{GL}_{2n}(\mathbb{A})\) and the metaplectic group \(\widetilde{\mathrm{Sp}_n}(\mathbb{A})\). As consequences, we expect a generalization of work of W. Kohnen [Math. Ann. 271, 237–268 (1985; Zbl 0542.10018)] and a verification of a conjecture of M. Furusawa and K. Martin [J. Number Theory 146, 150–170 (2015; Zbl 1366.11076)] characterizing \(\mathrm{GL}_{n}(K)\)-distinction of a cuspidal representation of \(\mathrm{GL}_{2n}(\mathbb{A})\).