$T_3$ ∧ 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 $T_{3 \frac{1}{2}}$ ).

The converse ( Realcompact ⇒ $T_3$ ∧ Lindelöf ) does not hold, as witnessed by these counterexamples:

Id	Name	$T_3$	Lindelöf	Realcompact
S107	Countable box product of reals Countable boolean product of reals, Box product topology on R^ω, Box product topology on R^omega

T3T_3T3​ ∧ Lindelöf ⇒ Realcompact