Lindelöf Tychonoff spaces and realcompactness in ZF

Ascertain whether, in ZF, every Tychonoff Lindelöf space is realcompact, or whether there exists a model of ZF containing a Tychonoff Lindelöf space that is not realcompact.

Background

In ZFC, all regular Lindelöf T1 spaces are realcompact. The authors prove in ZF that every zero-dimensional Lindelöf T1 space is N-compact and, under CMC, that every Tychonoff Lindelöf space is realcompact. The unresolved issue is whether ZF alone suffices to ensure realcompactness of Tychonoff Lindelöf spaces, or whether a counterexample exists in some model of ZF.