  We present a new partial order for directly forcing morasses to exist that
enjoys a significant homogeneity property. We then use this forcing in a
reverse Easton iteration to obtain an extension universe with morasses at every
regular uncountable cardinal, while preserving all n-superstrong (0<n<omega+1),
hyperstrong and 1-extendible cardinals. In the latter case, a preliminary
forcing to make the GCH hold is required. Our forcing yields morasses that
satisfy an extra property related to the homogeneity of the partial order; we
refer to them as mangroves and prove that their existence is equivalent to the
existence of morasses. Finally, we exhibit a partial order that forces
universal morasses to exist at every regular uncountable cardinal, and use this
to show that universal morasses are consistent with n-superstrong, hyperstrong,
and 1-extendible cardinals. This all contributes to the second author's outer
model programme, the aim of which is to show that L-like principles can hold in
outer models which nevertheless contain large cardinals.
