WCF - The HTTP request was forbidden with client authentication scheme 'Anonymous' - wcf

Question

I know this is a very common scenario, but I've still, after two days of searching, not found a solution to this problem.
I've got a WCF Service and a client (web site), using SSL and a client certificate.
Relevant service config section:
<system.serviceModel>
<bindings>
<basicHttpBinding>
<binding name="HOLBinding">
<security mode="Transport">
<transport clientCredentialType="Certificate" />
</security>
</binding>
</basicHttpBinding>
</bindings>
<behaviors>
<serviceBehaviors>
<behavior name="HOLServiceBehaviour">
<serviceCredentials>
<clientCertificate>
<authentication certificateValidationMode="PeerOrChainTrust" trustedStoreLocation="LocalMachine" />
</clientCertificate>
</serviceCredentials>
<serviceMetadata httpGetEnabled="false" httpsGetEnabled="false" />
<serviceDebug includeExceptionDetailInFaults="false" />
</behavior>
</serviceBehaviors>
<services>
<service name="HOL.Core.Service.HOLService" behaviorConfiguration="HOLServiceBehaviour">
<endpoint address="bh" bindingConfiguration="HOLBinding" binding="basicHttpBinding" contract="HOL.Core.Service.IHOLService" />
<endpoint address="wb" behaviorConfiguration="WebBehaviour" binding="webHttpBinding" contract="HOL.Core.Service.IHOLService" />
</service>
</services>
Relevant client service config:
<system.serviceModel>
<behaviors>
<endpointBehaviors>
<behavior name="HOLServiceEndpointBehaviour">
<clientCredentials>
<clientCertificate storeLocation="LocalMachine"
findValue="mythumbprint"
x509FindType="FindByThumbprint" storeName="TrustedPeople" />
<serviceCertificate>
<authentication certificateValidationMode="PeerOrChainTrust"/>
</serviceCertificate>
</clientCredentials>
</behavior>
</endpointBehaviors>
</behaviors>
<bindings>
<basicHttpBinding>
<binding name="BasicHttpBinding_IHOLService" maxBufferPoolSize="20000000" maxReceivedMessageSize="20000000">
<security mode="Transport">
<transport clientCredentialType="Certificate" />
</security>
</binding>
</basicHttpBinding>
</bindings>
<client>
<endpoint address="https://www.myhttpsite.co.uk/Service/HOLService.svc/bh" binding="basicHttpBinding"
bindingConfiguration="BasicHttpBinding_IHOLService" contract="HOLCoreService.IHOLService"
name="BasicHttpBinding_IHOLService" behaviorConfiguration="HOLServiceEndpointBehaviour">
</endpoint>
</client>
</system.serviceModel>
My certificate is being found, so that's not the problem (took me a day to fix that problem too!)
I believe the error is that the client calling the WCF Service simply isn't sending through the correct details to authenticate... but why?