  Consider a multitype coalescent process in which each block has a colour in
$\{1,\ldots,d\}$. Individual blocks may change colour, and some number of
blocks of various colours may merge to form a new block of some colour. We show
that if the law of a multitype coalescent process is invariant under
permutations of blocks of the same colour, has consistent Markovian
projections, and has asychronous mergers, then it is a multitype
$\Lambda$-coalescent: a process in which single blocks may change colour, two
blocks of like colour may merge to form a single block of that colour, or large
mergers across various colours happen at rates governed by a $d$-tuple of
measures on the unit cube $[0,1]^d$. We go on to identify when such processes
come down from infinity. Our framework generalises Pitman's celebrated
classification theorem for singletype coalescent processes, and provides a
unifying setting for numerous examples that have appeared in the literature
including the seed-bank model, the island model and the coalescent structure of
continuous-state branching processes.
