You probably could, but it probably wouldn't be the best way to achieve the co-owners goals in most cases in England and Wales and in other countries with a land registration system. It is more attractive in the U.S.
England and Wales v. U.S. Analysis Compared
The big factor that distinguishes England and Wales (and most but not all other Commonwealth countries) from the U.S., is that England and Wales has a system of title registration, in which certificates of title (possibly virtual these days rather than on a dead tree, I don't know, New Zealand recently made that transition), while almost all U.S. land records are maintained in one of three thousand or so separate county race-notice recording statute systems (which differ in fine details from state to state). The U.S. system is more flexible but also more prone to cases of title defects. The title registration system virtually eliminates the need for title insurance in commercial real estate transactions (i.e. as opposed to donative transfers of real estate, not as opposed to residential real estate).
In England and Wales
It is possible to achieve this goal and the approach suggested would rarely be the optimal solution.
The most straightforward way to do it would be to vest title not equally, but as an unequal tenancy in common with percentages stated on the face of the deed. But this might be a problem if the amount of contribution to the mortgage and capital improvements in the premises weren't known in advance when the parties took title.
Another solution which would achieve this end would be to vest title in an entity (including a possible general partnership or trust) which has provisions governing these matters. Entity ownership of some sort would also generally eliminate the partition rights of the co-owners, although a covenant to eliminate the right of partition could also be recorded in connection with the registered title without ending its 50-50 ownership character of record.
If the co-owners are married, title is a somewhat secondary matter and a premarital agreement (a.k.a. a prenuptial agreement) could probably achieve this end, at least in cases involving divorce or inheritance, although the marital rights established by the prenuptial or postnuptial agreement might be inchoate rather than vested and currently in force as a present interest.
With unmarried co-owners you could have a co-tenancy agreement (also called a tenants-in-common agreement and closely related to a joint ownership agreement) (or a domestic partnership agreement (also called a cohabitation agreement) which typically has a broader scope), but if that agreement were contractual, rather than reflected in the land registration as a percentage tenancy in common or as a covenant (which would be problematic in this case where the deed and the covenant contradict each other), the remedies would probably be limited if not in the land registry. Vis-a-vis the world, for example, in eminent domain proceedings or a probate proceeding at the death of one of the two co-owners, the co-owners would be treated as 50-50 owners.
A tenancy-in-common agreement included in the land registry would typically not dictate how the co-owners would manage the property, and instead would merely state their ownership rights. The other agreements would often contain management provisions for the co-owned property.
The contract would not create in rem rights in the property, unless it is recorded in the land registry (which a tenancy-in-common agreement often would be, effectively amending the 50-50 deed, but which the other forms of agreement generally would not be). But the parties would have a legal obligation to settle up between themselves following a liquidation event such as an eminent domain award or a sale of the property or a foreclosure, with in personam litigation between the co-owners, if necessary. In most circumstances, this would be equivalent, albeit a bit different procedurally. But, if a co-owner having an obligation under the contract to the other co-owner, were, for example, insolvent and there were other creditors of the insolvent co-owner with a higher priority for payment than the solvent co-owner of the property, the solvent co-owner of the property would lose out.
Any such agreement would have to be in a signed writing and not oral, as it would concern an interest in property.
I don't know how such a conflict between a 50-50 deed and a side contract would play out under the laws of England and Wales, for example, for purposes of taxes owed upon the sale of the property at a profit or a loss, or for purposes of determining someone's net worth for some financial accounting report in connection, for example, with a loan application in which the borrower is a limited liability entity and a co-owner is a guarantor of a loan.
In the U.S.
In the U.S., any document signed and notarized by the owners of the property or the affected parties, that is recorded in the real estate records of the county where the real estate is located, that adequately describes the property in question with a legal description, is given legal effect as a vested property right superseding any previous recorded document. Even if it isn't notarized, the document could also be supported by an affidavit or acknowledgement separate from the primary recorded document itself.
So, if there was a 50-50 tenancy in common deed of record, and then the co-owners recorded a subsequent tenancy-in-common or domestic partnership or premarital agreement, the recorded agreement would as a matter of property rights, modify the original deed. And, in most jurisdictions which have a "race-notice" recording rule, even if the agreement were not recorded, it would be effective to create a property right and would not simply be a side contractual agreement that is in personam only, as to anyone who had actual notice of the existence and terms of the agreement. So, this kind of solution is more attractive in the U.S. than in England and Wales or most other commonwealth jurisdictions.
Of course, an uneven tenancy-in-common deed, or an entity ownership would still be solutions that are available, and these options (possibly with a waiver of a right to partition) might be desirable in some circumstances.
In the U.S., some of the main downsides to entity ownership (all dependent upon both state and federal law) would be the loss of favorable capital gains tax rates, if the entity is a C corporation, the loss of the exclusion of gain on the sale of an owner occupied residence for capital gains tax purposes, the loss of a right to a homestead exemption from creditors, the loss of qualification for favorable mortgage program only available for owner occupied residences, the loss of favorable property tax treatment for owner occupied residences, and the loss of favorable treatment in determining eligibility for means tested government programs such as the Medicaid Long Term Care (i.e. nursing home) benefit. But real property in an entity is not automatically subject to execution on money judgments against the owners of the entity in their names.