Compelling necessity is a phrase that implies a vital and unavoidable need for something. To express the same idea, one might use the vocabulary like an urgent requirement, an imperative demand, or an inescapable imperative. Other synonyms could include a pressing obligation, a critical need, or a crucial demand. An urgent call, an unyielding necessity or an uncompromising requirement could also be used to describe the same thing. In some cases, one might use the phrase "unavoidable imperative," "paramount importance," or "dire straits" to convey the same sense of urgency. Ultimately, each of these phrases carries the same implication of something that is unavoidable and vitally important.