It's very difficult, it it's possible at all. Internet browsers do it
because they are in complete control of their interaction with the server.
If you want to do the same, you'll either have to do a bunch of very
sophisticated programming (have you ever written code that generates code?),
or you'll have to perform the interaction with the server yourself, and do
without the conveniences that Microsoft offers you.

In particular, you can create your own subclass of
System.Web.Services.Protocols.SoapHttpClientProtocol. You'll have to
override the GetWebRequest method to return an instance of HttpWebRequest
with the KeepAlive property set to true.

But your proxy classes will have to derive from your new subclass. This
means that you won't be able to use Add Web Reference or WSDL.EXE to create
your proxy classes.

Unless, that is, you write a SoapProtocolImporter which will generate proxy
classes based on your own subclass instead of the default
SoapHttpClientProtocol.

I've never done these things, since they are too complicated for most
purposes. It's always been better to let .NET decide whether or not to keep
a connection open, rather than worrying about it myself. Certainly, if you
have determined that there's either a measurable performance impact, or if
you need to do it in order to interact with a server which requires it, then
you have to do it.

And good luck to you. If it works, you should publish the solution.