When an household needs to change its home, a new house must be bought and the old one must be sold. In order to complete these two transactions, the household can adopt either a sequential or a simultaneous search strategy. In sequential strategies, it first buys (or sells) and only after tries to sell (or buy), to avoid either being homeless or holding two houses, respectively. In the simultaneous strategy, the household tries to buy and sell simultaneously. If the household adopts the simultaneous strategy, it can reduce its search costs, but becomes exposed to the risk of becoming a homeless renter or the owner of two houses. The literature generally considers only the sequential search strategy. However, we show in this article that the simultaneous strategy is (i) generally welfare improving for households, (ii) sometimes the sole equilibrium strategy, and (iii) at the origin of price dispersion on the housing market.