Is it possible to pair TCP and UDP streams?

What I am looking for is a a way to instruct the load balancer to locate a key identifier in the UDP packet, use it to identify an existing TCP stream and direct the UDP packet to the same server.

your understanding is absolutely correct. locating key identifier is mandatory for persistence. match across comes in after that.

e.g.

 configuration

root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm virtual bar1
ltm virtual bar1 {
    destination 172.28.24.10:8
    ip-protocol tcp
    mask 255.255.255.255
    persist {
        myuie {
            default yes
        }
    }
    pool foo1
    profiles {
        tcp { }
    }
    rules {
        qux1
    }
    source 0.0.0.0/0
    source-address-translation {
        type automap
    }
    vs-index 10
}
root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm pool foo1
ltm pool foo1 {
    members {
        200.200.200.101:7 {
            address 200.200.200.101
        }
        200.200.200.111:7 {
            address 200.200.200.111
        }
    }
}
root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm persistence universal myuie
ltm persistence universal myuie {
    app-service none
    defaults-from universal
    match-across-virtuals enabled
}
root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm rule qux1
ltm rule qux1 {
    when CLIENT_ACCEPTED {
  TCP::collect
}
when CLIENT_DATA {
  set key [TCP::payload]
  persist uie [TCP::payload]
  TCP::release
}
when SERVER_CONNECTED {
  log local0. "key=$key server=[IP::server_addr]:[TCP::server_port]"
}
}

root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm virtual bar2
ltm virtual bar2 {
    destination 172.28.24.100:88
    ip-protocol udp
    mask 255.255.255.255
    persist {
        myuie {
            default yes
        }
    }
    pool foo2
    profiles {
        udp { }
    }
    rules {
        qux2
    }
    source 0.0.0.0/0
    source-address-translation {
        type automap
    }
    vs-index 11
}
root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm pool foo2
ltm pool foo2 {
    members {
        200.200.200.101:77 {
            address 200.200.200.101
        }
        200.200.200.111:77 {
            address 200.200.200.111
        }
    }
}
root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm rule qux2
ltm rule qux2 {
    when CLIENT_ACCEPTED {
  set key [UDP::payload]
  persist uie [UDP::payload]
}
when SERVER_CONNECTED {
  log local0. "key=$key server=[IP::server_addr]:[UDP::server_port]"
}
}

 test

root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) show ltm persistence persist-records all-properties
Sys::Persistent Connections
universal - 172.28.24.10:8 - 200.200.200.101:7
----------------------------------------------
  TMM           1
  Mode          universal
  Value         a

  Age (sec.)    8
  Virtual Name  /Common/bar1
  Virtual Addr  172.28.24.10:8
  Node Addr     200.200.200.101:7
  Pool Name     /Common/foo1
  Client Addr   172.28.24.1
  Owner entry

universal - 172.28.24.10:8 - 200.200.200.111:7
----------------------------------------------
  TMM           1
  Mode          universal
  Value         c

  Age (sec.)    32
  Virtual Name  /Common/bar1
  Virtual Addr  172.28.24.10:8
  Node Addr     200.200.200.111:7
  Pool Name     /Common/foo1
  Client Addr   172.28.24.1
  Owner entry

universal - 172.28.24.10:8 - 200.200.200.111:7
----------------------------------------------
  TMM           0
  Mode          universal
  Value         b

  Age (sec.)    35
  Virtual Name  /Common/bar1
  Virtual Addr  172.28.24.10:8
  Node Addr     200.200.200.111:7
  Pool Name     /Common/foo1
  Client Addr   172.28.24.1
  Owner entry

Total records returned: 3

[root@ve11a:Active:In Sync] config  cat /var/log/ltm
Dec 14 13:05:34 ve11a info tmm[14890]: Rule /Common/qux1 : key=a  server=200.200.200.101:7
Dec 14 13:05:37 ve11a info tmm1[14890]: Rule /Common/qux1 : key=b  server=200.200.200.111:7
Dec 14 13:05:39 ve11a info tmm[14890]: Rule /Common/qux1 : key=c  server=200.200.200.111:7
Dec 14 13:05:45 ve11a info tmm1[14890]: Rule /Common/qux2 : key=c  server=200.200.200.111:77
Dec 14 13:05:50 ve11a info tmm[14890]: Rule /Common/qux2 : key=b  server=200.200.200.111:77
Dec 14 13:05:53 ve11a info tmm1[14890]: Rule /Common/qux2 : key=c  server=200.200.200.111:77
Dec 14 13:06:17 ve11a info tmm[14890]: Rule /Common/qux2 : key=a  server=200.200.200.101:77