  Given a Schauder basic sequence $(x_k)$ in a Banach lattice, we say that
$(x_k)$ is bibasic if the expansion of every vector in $[x_k]$ converges not
only in norm, but also in order. We prove that, in this definition, order
convergence may be replaced with uniform convergence, with order boundedness of
the partial sums, or with norm boundedness of finite suprema of the partial
sums.
  The results in this paper extend and unify those from the pioneering paper
"Order Schauder bases in Banach lattices" by A.Gumenchuk et al. In particular,
we are able to characterize bibasic sequences in terms of the bibasis
inequality, a result they obtained under certain additional assumptions.
  We then embark on a deeper study of their properties. We show that they are
independent of ambient space, stable under small perturbations, and preserved
under sequentially uniformly continuous norm isomorphic embeddings. We consider
several special kinds of bibasic sequences, including permutable sequences,
i.e., sequences for which every permutation is bibasic, and absolute sequences,
i.e., sequences where expansions remain convergent after we replace every term
with its modulus. We provide several equivalent characterizations of absolute
sequences, showing how they relate to bibases and to further modifications of
the basis inequality.
  We further consider bibasic sequences with unique order expansions. We show
that this property does generally depend on ambient space, but not for the
inclusion of $c_0$ into $\ell_\infty$. We also show that small perturbations of
bibases with unique order expansions have unique order expansions, but this is
not true if "bibases" is replaced with "bibasic sequences".
  Finally, we consider uo-bibasic sequences, which are obtained by replacing
order convergence with uo-convergence in the definition of a bibasic sequence.
We show that such sequences are very common.
