No it's not possible today, because neither CSCO or JNPR have implemented LDP to distribute labels for IPV6 prefixes.
In my opinion people have too sentimental view on this matter, thinking IPV6-only is value in itself, it's not. You should differentiate services you offer from your control-plane used to offer those.
There is no particular reason to have dual-stack control-plane, it's just additional complexity. Personally, I'd be happy to run IPv4 + 6PE until in some far future I'll fork-lift or build new network with IPv6 + 4PE.
Running MPLS for IPv4 and native IPv6 is probably the silliest thing you could do, as you'll remove all the scaling, TE and convergence benefits you get from MPLS.
Segment Routing/SRING which uses IGP for labels instead of LDP will likely happen before LDP for IPv6 prefixes happen. So if you really for some strange reason want to have both control-planes that's your most likely source to get it soon.