In a multi-region web app, is it common to have the main user database in a single region or to replicate it across multiple regions and lose consistency?
I don't think it's possible to answer your question definitively without considering your application requirements, projected throughput and trade-offs you are willing to make.
I'm assuming you are looking at having a number of SQL DB servers working in Multi-master replication scenario.
You can potentially shard your data in such a way that even though all replicas would have full data set - each regional app server would only need access to a subset that's always located in the same region. Usefulness of such approach would very much depend on what data you have and what you do with them.
If you don't have issue with using public cloud, you could look at either Azure Cosmos DB or AWS Aurora - both offer multi-region DBs with varying consistency settings. Meaning that albeit not immediate, you could get decent latency guarantees that might be enough for your use case.
I don't think it's possible to have 100% consistent cross-data center distributed system. So if you absolutely need strict consistency, you probably are stuck with having one DB. Given you specifically mention user database, it might be that all this complexity is not worth the effort as you could get away by scaling it up a bit and designing your user security in a way that minimises trips to DB and, for example, employs signed JWT tokens for subsequent requests.