A home sale contingency is a provision in a real estate purchase agreement that makes the buyer's obligation to complete the purchase conditional on the sale of a home the buyer already owns. It is a creature of the contract rather than of federal law: no federal mortgage regulation prescribes its wording, and what it covers, how long it runs and what happens on termination are set by the form the parties sign and by state contract law.
Federal regulation reaches the situation from one side only, by describing the financing route a buyer takes when a seller will not accept the condition. Regulation Z excludes from its ability-to-repay determination "a temporary or 'bridge' loan with a term of 12 months or less, such as a loan to finance the purchase of a new dwelling where the consumer plans to sell a current dwelling within 12 months" (12 CFR 1026.43(a)(3)(ii)). That sentence is a fair statement of the problem: the buyer's money is in one house and the purchase is of another. A bridge loan solves it with debt; this clause solves it with a condition.