I will introduce a family of marked determinantal point processes and associated conditional ensembles, in which information about a random part of a point configuration is encoded. Special cases of these conditional ensembles appear in the Its-Izergin-Korepin-Slavnov method and in the study of number rigidity. I will discuss general properties of these ensembles, and show how they lead to a strengthened notion of number rigidity for determinantal point processes induced by a certain class of orthogonal projections, including the sine, Airy, and Bessel point processes. The talk will be based on joint work with Gabriel Glesner.