In the context of search problems, a pseudodeterministic algorithm is one which uses randomness but appears to be deterministic in its external behavior: it outputs the same answer when it is run on any one input multiple times. We introduce the relaxed notion of computationally-secure pseudodeterminism, which we call faux determinism. A faux-deterministic algorithm must appear to be deterministic to any external adversary with bounded resources. In other words, an algorithm is faux-deterministic if the task of "finding an input on which it fails to deterministically solve the search problem" is computationally intractable. In the black-box setting, we show that every verifiable search problem which has a randomized algorithm also has a faux-deterministic algorithm; this is not true of pseudodeterministic algorithms, which do not always exist. We highlight two reasons why this faux derandomization result is interesting: first, due to the computational indistinguishability, faux-deterministic algorithms may be used in place of pseudodeterministic ones in cryptographic settings. Second, from a conceptual point of view, our result helps explain why pseudodeterministic lower bounds are often hard to prove: such lower bounds must be inherently nonconstructive, since within FZPP∩FNP, there is no efficient black-box algorithm for finding a hard input for a given faux-deterministic algorithm.