$T_3$ ∧ Lindelöf ⇒ Realcompact

See Theorem 3.8.2 of zbMATH 0684.54001 (where the Lindelöf property assumes $T_3$).

Also Theorem 8.2 of DOI 10.1007/978-1-4615-7819-2 (where all spaces are assumed $T_{3 \frac{1}{2}}$).