Semi-compensatory models show promise in representing two-stage choice processes by integrating choice set formation within discrete choice models. However, they are subject to simplifying assumptions that limit their application. The current study proposes a novel semi-compensatory model that alleviates these simplifying assumptions concerning (i) the number of alternatives, (ii) the representation of choice set formation, and (iii) the error structure. The proposed semi-compensatory model represents a sequence of choice set formation based on the conjunctive heuristic with correlated thresholds, and utility-based choice accommodating alternatively nested substitution patterns across the alternatives and random taste variation across the population. The proposed model is applied to off-campus rental apartment choice of students. Results show (i) the estimated model for a universal realm of 200 alternatives and 41 choice sets, (ii) the threshold representation as a function of individual characteristics, and (iii) the feasibility and importance of introducing a flexible error structure into semi-compensatory models.