  This paper investigates Everpresent $\Lambda$, a stochastic dark energy model
motivated by causal set theory and unimodular gravity, and confronts it with
two key observational data sets, Supernova Ia (SN Ia) and Cosmic Microwave
Background (CMB) data. A key feature of this model is that $\Lambda$ fluctuates
over time and on average the magnitude of its fluctuations is of the order of
the dominant energy density (be it radiation or matter) for the given epoch. In
particular, we focus on a phenomenological implementation of Everpresent
$\Lambda$ known as Model 1. The random fluctuations in Everpresent $\Lambda$
realizations are generated using seed numbers, and we find that for a small
fraction of seeds Model 1 is capable of producing realizations that fit SN Ia
data better than $\Lambda$CDM. We further investigate what features distinguish
these realizations from the more general behaviour, and find that the "good''
realizations have relatively small fluctuations at low redshifts ($z<1.5$),
which do not closely track the matter density. We find that Model 1 struggles
to improve on $\Lambda$CDM at describing the CMB data. However, by suppressing
the values of $\Lambda$ near the last scattering surface, as suggested in
arXiv:1703.06265, we find a large improvement in the best fit of the model,
though still with a $\chi^2$ value much larger than that of $\Lambda$CDM. We
also study the allowed variation of the dark energy density by the CMB
constraints in a more model-independent manner, and find that some variation
(especially prior to recombination) is possible and in fact can lead to
improvement over $\Lambda$CDM and reduce the Hubble tension, in line with some
early dark energy proposals. However, for the kinds of variations considered,
the favoured fluctuations are smaller in magnitude than is typical in current
Everpresent $\Lambda$ models.
