We show that nearly half of all binary black hole (BBH) mergers dynamically assembled in globular clusters have measurable eccentricities (e > 0.01) in the LISA band (10^-2 Hz), when General Relativistic corrections are properly included in the N-body evolution. If only Newtonian gravity is included, the derived fraction of eccentric LISA sources is significantly lower, which explains why recent studies all have greatly underestimated this fraction. Our findings have major implications for how to observationally distinguish between BBH formation channels using eccentricity with LISA, which is one of the key science goals of the mission. We illustrate that the relatively large population of eccentric LISA sources reported here originates from BBHs that merge between hardening binary-single interactions inside their globular cluster. These results indicate a bright future for using LISA to probe the origin of BBH mergers.