  In this paper we present a surprisingly general extension of the main result
of a paper that appeared in this journal: I. Montes et al., Sklar's theorem in
an imprecise setting, Fuzzy Sets and Systems, 278 (2015), 48--66. The main
tools we develop in order to do so are: (1) a theory on quasi-distributions
based on an idea presented in a paper by R. Nelsen with collaborators; (2)
starting from what is called (bivariate) $p$-box in the above mentioned paper
we propose some new techniques based on what we call restricted (bivariate)
$p$-box; and (3) a substantial extension of a theory on coherent imprecise
copulas developed by M. Omladi\v{c} and N. Stopar in a previous paper in order
to handle coherence of restricted (bivariate) $p$-boxes. A side result of ours
of possibly even greater importance is the following: Every bivariate
distribution whether obtained on a usual $\sigma$-additive probability space or
on an additive space can be obtained as a copula of its margins meaning that
its possible extraordinariness depends solely on its margins. This might
indicate that copulas are a stronger probability concept than probability
itself.
